6-4 Rectangles

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FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER: 6-4 Rectangles

FARMING An X-brace on a rectangular barn 3. door is both decorative and functional. It helps to prevent the door from warping over time. SOLUTION: The diagonals of a rectangle are congruent and If feet, PS = 7 feet, and bisect each other. So, is an isosceles triangle. , find each measure. Then, By the Exterior Angle Theorem,

Therefore,

ANSWER: 33.5

4. SOLUTION:

1. QR The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. SOLUTION: Then, The opposite sides of a rectangle are parallel and By the Vertical Angle Theorem, congruent. Therefore, QR = PS = 7 ft. ANSWER: Therefore, 7 ft ANSWER: 2. SQ 56.5 SOLUTION: ALGEBRA Quadrilateral DEFG is a rectangle. The diagonals of a rectangle bisect each other. So,

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG. ANSWER:

Use the value of x to find EG. 3. EG = 6 + 5 = 11

SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 11 bisect each other. So, is an isosceles triangle. Then, 6. If , By the Exterior Angle Theorem,

find . eSolutions Manual - Powered by Cognero SOLUTION: Page 1 Therefore, All four angles of a rectangle are right angles. So,

ANSWER: 33.5

SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. ANSWER: Then, 51 By the Vertical Angle Theorem, 7. PROOF If ABDE is a rectangle and , Therefore, prove that .

ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

SOLUTION: You need to walk through the proof step by step.

5. If FD = 3x – 7 and EG = x + 5, find EG. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and SOLUTION: . You need to prove . Use the The diagonals of a rectangle are congruent to each properties that you have learned about rectangles to other. So, FD = EG. walk through the proof.

Given: ABDE is a rectangle; Prove: Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 11

6. If , find . Statements(Reasons) 1. ABDE is a rectangle; . SOLUTION: (Given) All four angles of a rectangle are right angles. So, 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: ANSWER: 51 Given: ABDE is a rectangle; 7. PROOF If ABDE is a rectangle and , Prove: prove that .

Statements(Reasons) 1. ABDE is a rectangle; . SOLUTION: (Given)

You need to walk through the proof step by step. 2. ABDE is a parallelogram. (Def. of

Look over what you are given and what you need to rectangle) prove. You are given ABDE is a rectangle and 3. (Opp. sides of a .) . You need to prove . Use the 4. are right angles.(Def. of rectangle) properties that you have learned about rectangles to 5. (All rt .) walk through the proof. 6. (SAS)

7. (CPCTC) Given: ABDE is a rectangle; Prove: COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) Use the slope formula to find the slope of the sides of 7. (CPCTC) the quadrilateral. ANSWER: Given: ABDE is a rectangle; Prove:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. Statements(Reasons) 1. ABDE is a rectangle; . ANSWER: (Given) 2. ABDE is a parallelogram. (Def. of No; slope of , slope of , slope of rectangle) 3. (Opp. sides of a .) , and slope of . Slope of 4. are right angles.(Def. of rectangle) slope of , and slope of slope of 5. (All rt .) , so WXYZ is not a parallelogram. Therefore, 6. (SAS) WXYZ is not a rectangle. 7. (CPCTC)

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER: so ABCD is a No; slope of , slope of , slope of parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the , and slope of . Slope of lengths of the diagonals. slope of , and slope of slope of , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

10. BC SOLUTION: so ABCD is a The opposite sides of a rectangle are parallel and

parallelogram. congruent. Therefore, BC = AD = 2 ft. A parallelogram is a rectangle if the diagonals are ANSWER: congruent. Use the Distance formula to find the 2 ft lengths of the diagonals.

11. DB So the diagonals are congruent. Thus, ABCD is a SOLUTION: rectangle. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean ANSWER: 2 2 2 Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a Theorem, DB = BC + DC . BC = AD = 2 parallelogram. , so the diagonals are DC = AB = 6 congruent. Thus, ABCD is a rectangle.

ANSWER: 6.3 ft

12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. FENCING X-braces are also used to provide Then, support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure. By the Vertical Angle Theorem,

ANSWER: 50

13. SOLUTION: The diagonals of a rectangle are congruent and 10. BC bisect each other. So, is an isosceles SOLUTION: triangle. The opposite sides of a rectangle are parallel and Then, congruent. Therefore, BC = AD = 2 ft. All four angles of a rectangle are right angles.

ANSWER: 2 ft

11. DB ANSWER: 25 SOLUTION: All four angles of a rectangle are right angles. So, CCSS REGULARITY Quadrilateral WXYZ is a is a right triangle. By the Pythagorean rectangle. 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

14. If ZY = 2x + 3 and WX = x + 4, find WX. ANSWER: SOLUTION: 6.3 ft The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 12. 2x + 3 = x + 4 x = 1 SOLUTION: Use the value of x to find WX. The diagonals of a rectangle are congruent and WX = 1 + 4 = 5. bisect each other. So, is an isosceles triangle. ANSWER: Then, 5

15. If PY = 3x – 5 and WP = 2x + 11, find ZP.

By the Vertical Angle Theorem, SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP. ANSWER: 50 The diagonals of a rectangle are congruent and 13. bisect each other. So, is an isosceles triangle. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, ANSWER: All four angles of a rectangle are right angles. 43

16. If , find . SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 25

CCSS REGULARITY Quadrilateral WXYZ is a rectangle.

ANSWER: 14. If ZY = 2x + 3 and WX = x + 4, find WX. 39 SOLUTION: 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. The opposite sides of a rectangle are parallel and SOLUTION: congruent. So, ZY = WX. 2x + 3 = x + 4 The diagonals of a rectangle are congruent and

x = 1 bisect each other. So, ZP = PY. Use the value of x to find WX. WX = 1 + 4 = 5.

ANSWER: Then ZP = 4(7) – 9 = 19. 5 Therefore, ZX = 2(ZP) = 38.

ANSWER: 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. 38 SOLUTION: The diagonals of a rectangle bisect each other. So, 18. If , find PY = WP. .

SOLUTION: All four angles of a rectangle are right angles. So, The diagonals of a rectangle are congruent and

bisect each other. So, is an isosceles triangle.

ANSWER: 43 ANSWER: 48 16. If , find . 19. If , find . SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. Then,

Therefore, Since is a ANSWER: transversal of parallel sides and , alternate 39 interior angles WZX and ZXY are congruent. So, 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. = = 46. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 46 bisect each other. So, ZP = PY. PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. Prove: Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38.

ANSWER:

38 SOLUTION: 18. If , find You need to walk through the proof step by step. . Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. SOLUTION: You need to prove . Use the All four angles of a rectangle are right angles. So, properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) ANSWER: 2. ABCD is a parallelogram. (Def. of rectangle) 48 3. (Opp. sides of a .) 19. If , find 4. (Refl. Prop.) . 5. (Diagonals of a rectangle are .) SOLUTION: 6. (SSS) All four angles of a rectangle are right angles. So, ANSWER: is a right triangle. Then, Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) Therefore, Since is a 4. (Refl. Prop.) transversal of parallel sides and , alternate 5. (Diagonals of a rectangle are .) interior angles WZX and ZXY are congruent. So, 6. (SSS) = = 46. 21. Given: QTVW is a rectangle. ANSWER:

46 PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. Prove:

Prove: SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to SOLUTION: prove. Here, you are given QTVW is a rectangle and You need to walk through the proof step by step. . You need to prove . Use Look over what you are given and what you need to the properties that you have learned about rectangles prove. Here, you are given ABCD is a rectangle. to walk through the proof. You need to prove . Use the properties that you have learned about rectangles to Given: QTVW is a rectangle. walk through the proof.

Given: ABCD is a rectangle. Prove: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) Prove:

2. ABCD is a parallelogram. (Def. of rectangle) Proof: 3. (Opp. sides of a .) Statements (Reasons) 4. (Refl. Prop.) 1. QTVW is a rectangle; .(Given) 5. (Diagonals of a rectangle are .) 2. QTVW is a parallelogram. (Def. of rectangle) 6. (SSS) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) ANSWER: 5. (All rt .) Proof: Statements (Reasons) 6. QR = ST (Def. of segs.) 1. ABCD is a rectangle. (Given) 7. (Refl. Prop.) 2. ABCD is a parallelogram. (Def. of rectangle) 8. RS = RS (Def. of segs.) 3. (Opp. sides of a .) 9. QR + RS = RS + ST (Add. prop.) 4. (Refl. Prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 5. (Diagonals of a rectangle are .) 6. (SSS) 12. (Def. of segs.) 13. (SAS)

21. Given: QTVW is a rectangle. ANSWER: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle)

3. (Opp sides of a .) Prove: 4. are right angles.(Def. of rectangle) 5. (All rt .) SOLUTION: 6. QR = ST (Def. of segs.) You need to walk through the proof step by step. Look over what you are given and what you need to 7. (Refl. Prop.) prove. Here, you are given QTVW is a rectangle and 8. RS = RS (Def. of segs.) . You need to prove . Use 9. QR + RS = RS + ST (Add. prop.) the properties that you have learned about rectangles 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) to walk through the proof. 11. QS = RT (Subst.) 12. (Def. of segs.) Given: QTVW is a rectangle. 13. (SAS)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula Prove: SOLUTION: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.)

9. QR + RS = RS + ST (Add. prop.) Use the slope formula to find the slope of the sides of 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) the quadrilateral. 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) The slopes of each pair of opposite sides are equal. 4. are right angles.(Def. of rectangle) So, the two pairs of opposite sides are parallel. 5. (All rt .) Therefore, the quadrilateral WXYZ is a parallelogram. The products of the slopes of the adjacent sides are 6. QR = ST (Def. of segs.) –1. So, any two adjacent sides are perpendicular to 7. (Refl. Prop.) each other. That is, all the four angles are right 8. RS = RS (Def. of segs.) angles. Therefore, WXYZ is a rectangle. 9. QR + RS = RS + ST (Add. prop.) ANSWER: 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) Yes; slope of slope of , slope 12. (Def. of segs.) 13. (SAS) of slope of . So WXYZ is a parallelogram. The product of the slopes of COORDINATE GEOMETRY Graph each consecutive sides is –1, so the consecutive sides are quadrilateral with the given vertices. Determine perpendicular and form right angles. Thus, WXYZ is a whether the figure is a rectangle. Justify your rectangle. answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION:

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula

Use the slope formula to find the slope of the sides of SOLUTION: the quadrilateral.

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. First, Use the Distance formula to find the lengths of The products of the slopes of the adjacent sides are the sides of the quadrilateral. –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle.

ANSWER: so JKLM is a Yes; slope of slope of , slope parallelogram. A parallelogram is a rectangle if the diagonals are of slope of . So WXYZ is a congruent. Use the Distance formula to find the parallelogram. The product of the slopes of lengths of the diagonals. consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle. So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so JKLM is a First, use the Distance Formula to find the lengths of parallelogram. the sides of the quadrilateral. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

so QRST is a So the diagonals are not congruent. Thus, JKLM is parallelogram. not a rectangle. A parallelogram is a rectangle if the diagonals are ANSWER: congruent. Use the Distance Formula to find the lengths of the diagonals. No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a So the diagonals are congruent. Thus, QRST is a rectangle. rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so QRST is a First, use the slope formula to find the lengths of the parallelogram. sides of the quadrilateral. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. parallelogram. so the diagonals are Therefore, the quadrilateral GHJK is a congruent. QRST is a rectangle. parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle.

ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle. 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

First, use the slope formula to find the lengths of the sides of the quadrilateral.

26. SOLUTION: All four angles of a rectangle are right angles. So,

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. ANSWER: Therefore, the quadrilateral GHJK is a parallelogram. 50 None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. 27. Therefore, GHJK is not a rectangle. SOLUTION: ANSWER: The measures of angles 2 and 3 are equal as they are alternate interior angles. No; slope of slope of and slope of Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 slope of . So, GHJK is a and 8 is an isosceles triangle. So, m 7 = m 3. parallelogram. The product of the slopes of ∠ ∠ consecutive sides ≠ –1, so the consecutive sides are Therefore, m∠7 = m∠2 = 40.

not perpendicular. Thus, GHJK is not a rectangle. ANSWER: 40

28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. Quadrilateral ABCD is a rectangle. Find each SOLUTION: measure if . All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. 26. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 All four angles of a rectangle are right angles. So, and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m 5 = 180 – (50 + 50) = 80. ∠ ANSWER: 80 ANSWER: 50 30. SOLUTION:

27. All four angles of a rectangle are right angles. So, SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and The measures of angles 1 and 4 are equal as they bisect each other, the triangle with the angles 3, 7 are alternate interior angles. and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠4 = m∠1 = 50. Therefore, m∠7 = m∠2 = 40. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 ANSWER: and 6 is an isosceles triangle. Therefore, m∠6 = 40 m∠4 = 50. 28. ANSWER: SOLUTION: 50 The measures of angles 2 and 3 are equal as they 31. are alternate interior angles. Therefore, m∠3 = m∠2 = 40. SOLUTION: The measures of angles 2 and 3 are equal as they ANSWER: are alternate interior angles. 40 Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 29. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. SOLUTION: The sum of the three angles of a triangle is 180. All four angles of a rectangle are right angles. So, Therefore, m∠8 = 180 – (40 + 40) = 100.

ANSWER: 100

The measures of angles 1 and 4 are equal as they 32. CCSS MODELING Jody is building a new are alternate interior angles. bookshelf using wood and metal supports like the one Therefore, m∠4 = m∠1 = 50. shown. To what length should she cut the metal Since the diagonals of a rectangle are congruent and supports in order for the bookshelf to be square, bisect each other, the triangle with the angles 4, 5 which means that the angles formed by the shelves and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. and the vertical supports are all right angles? Explain The sum of the three angles of a triangle is 180. your reasoning. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: 80

30. SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we The measures of angles 1 and 4 are equal as they know the length of the bookshelves and the distance are alternate interior angles. between them, use the Pythagorean Theorem to find Therefore, m∠4 = m∠1 = 50. the lengths of the metal supports. Since the diagonals of a rectangle are congruent and Let x be the length of each support. Express 3 feet bisect each other, the triangle with the angles 4, 5 as 36 inches. and 6 is an isosceles triangle. Therefore, m∠6 = m∠4 = 50. ANSWER: 50 Therefore, the metal supports should each be 39 31. inches long or 3 feet 3 inches long.

SOLUTION: ANSWER: The measures of angles 2 and 3 are equal as they 3 ft 3 in.; Sample answer: In order for the bookshelf are alternate interior angles. to be square, the lengths of all of the metal supports Since the diagonals of a rectangle are congruent and should be equal. Since I knew the length of the bisect each other, the triangle with the angles 3, 7 bookshelves and the distance between them, I used and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. the Pythagorean Theorem to find the lengths of the The sum of the three angles of a triangle is 180. metal supports. Therefore, m∠8 = 180 – (40 + 40) = 100. PROOF Write a two-column proof. ANSWER: 33. Theorem 6.13 100 SOLUTION: 32. CCSS MODELING Jody is building a new You need to walk through the proof step by step. bookshelf using wood and metal supports like the one Look over what you are given and what you need to shown. To what length should she cut the metal prove. Here, you are given WXYZ is a rectangle with supports in order for the bookshelf to be square, diagonals . You need to prove which means that the angles formed by the shelves . Use the properties that you have learned and the vertical supports are all right angles? Explain about rectangles to walk through the proof. your reasoning.

Given: WXYZ is a rectangle with diagonals . Prove:

SOLUTION: Proof: In order for the bookshelf to be square, the lengths of 1. WXYZ is a rectangle with diagonals . all of the metal supports should be equal. Since we

know the length of the bookshelves and the distance (Given) between them, use the Pythagorean Theorem to find 2. (Opp. sides of a .) the lengths of the metal supports. 3. (Refl. Prop.) Let x be the length of each support. Express 3 feet 4. are right angles. (Def. of as 36 inches. rectangle) 5. (All right .) 6. (SAS) 7. (CPCTC)

Therefore, the metal supports should each be 39 ANSWER: inches long or 3 feet 3 inches long. Given: WXYZ is a rectangle with diagonals

ANSWER: . 3 ft 3 in.; Sample answer: In order for the bookshelf Prove: to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the metal supports. Proof: PROOF Write a two-column proof. 1. WXYZ is a rectangle with diagonals . 33. Theorem 6.13 (Given) SOLUTION: 2. (Opp. sides of a .) You need to walk through the proof step by step. Look over what you are given and what you need to 3. (Refl. Prop.) prove. Here, you are given WXYZ is a rectangle with 4. are right angles. (Def. of diagonals . You need to prove rectangle) 5. (All right .) . Use the properties that you have learned 6. (SAS) about rectangles to walk through the proof. 7. (CPCTC) Given: WXYZ is a rectangle with diagonals . 34. Theorem 6.14

Prove: SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given . You need to

Proof: prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk 1. WXYZ is a rectangle with diagonals . through the proof. (Given)

2. (Opp. sides of a .) Given: 3. (Refl. Prop.) Prove: WXYZ is a rectangle. 4. are right angles. (Def. of rectangle) 5. (All right .) 6. (SAS)

7. (CPCTC) Proof: ANSWER: 1. (Given) Given: WXYZ is a rectangle with diagonals 2. (SSS) . 3. (CPCTC) Prove: 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) Proof: 8. are right angles. (If 1. WXYZ is a rectangle with diagonals . 2 and suppl., each is a rt. .) (Given) 9. are right angles. (If a has 1 2. (Opp. sides of a .) rt. , it has 4 rt. .) 3. (Refl. Prop.) 10. WXYZ is a rectangle. (Def. of rectangle) 4. are right angles. (Def. of ANSWER: rectangle) 5. (All right .) Given: 6. (SAS) Prove: WXYZ is a rectangle. 7. (CPCTC)

34. Theorem 6.14 SOLUTION:

You need to walk through the proof step by step. Proof: Look over what you are given and what you need to 1. (Given) prove. Here, you are given 2. (SSS) . You need to 3. (CPCTC) prove that WXYZ is a rectangle.. Use the properties 4. (Def. of ) that you have learned about rectangles to walk 5. WXYZ is a parallelogram. (If both pairs of opp.

through the proof. sides are , then quad. is .)

6. are supplementary. (Cons. Given: are suppl.) Prove: WXYZ is a rectangle. 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) Proof: 10. WXYZ is a rectangle. (Def. of rectangle) 1. (Given) PROOF Write a paragraph proof of each 2. (SSS) statement. 3. (CPCTC) 35. If a parallelogram has one right angle, then it is a 4. (Def. of ) rectangle. 5. WXYZ is a parallelogram. (If both pairs of opp. SOLUTION: sides are , then quad. is .) You need to walk through the proof step by step. 6. are supplementary. (Cons. Look over what you are given and what you need to are suppl.) prove. Here, you are given a parallelogram with one 7. (Def of suppl.) right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. 8. are right angles. (If Then use the properties of parallelograms to walk 2 and suppl., each is a rt. .) through the proof. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

ANSWER: Given: Prove: WXYZ is a rectangle. ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle.

ANSWER: Proof: 1. (Given) 2. (SSS) 3. (CPCTC) 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. ABCD is a parallelogram, and is a right angle. sides are , then quad. is .) Since ABCD is a parallelogram and has one right 6. are supplementary. (Cons. angle, then it has four right angles. So by the are suppl.) definition of a rectangle, ABCD is a rectangle. 7. (Def of suppl.) 8. are right angles. (If 36. If a quadrilateral has four right angles, then it is a rectangle. 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 SOLUTION: rt. , it has 4 rt. .) You need to walk through the proof step by step. 10. WXYZ is a rectangle. (Def. of rectangle) Look over what you are given and what you need to prove. Here, you are given a quadrilateral with four PROOF Write a paragraph proof of each right angles. You need to prove that it is a rectangle. statement. Let ABCD be a quadrilateral with four right angles. 35. If a parallelogram has one right angle, then it is a Then use the properties of parallelograms and rectangle. rectangles to walk through the proof.

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk ABCD is a quadrilateral with four right angles. through the proof. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

ANSWER:

37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the ABCD is a parallelogram, and is a right angle. construction. Since ABCD is a parallelogram and has one right SOLUTION: angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. Sample answer: Step 1: Graph a line and place points P and Q on the line. 36. If a quadrilateral has four right angles, then it is a rectangle. Step 2: Place the compass on point P. Using the SOLUTION: same compass setting, draw an arc to the left and You need to walk through the proof step by step. right of P that intersects . Look over what you are given and what you need to prove. Here, you are given a quadrilateral with four Step 3: Open the compass to a setting greater than right angles. You need to prove that it is a rectangle. the distance from P to either point of intersection. Let ABCD be a quadrilateral with four right angles. Place the compass on each point of intersection and Then use the properties of parallelograms and rectangles to walk through the proof. draw arcs that intersect above .

Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3.

Step 5: Repeat steps 2 and 3 using point Q.

ANSWER: Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw . ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. SOLUTION: Sample answer: Since and , the measure of Step 1: Graph a line and place points P and Q on the angles P and Q is 90 degrees. Lines that are line. perpendicular to the same line are parallel, so

. The same compass setting was used to Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and right of P that intersects . congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Step 3: Open the compass to a setting greater than Thus, PRSQ is a rectangle. the distance from P to either point of intersection. Place the compass on each point of intersection and draw arcs that intersect above . ANSWER:

Step 4: Draw line a through point P and the Sample answer: Since and , the intersection of the arcs drawn in step 3. measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 5: Repeat steps 2 and 3 using point Q. . The same compass setting was used to

Step 6: Draw line b through point Q and the locate points R and S, so . If one pair of intersection of the arcs drawn in step 5. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 7: Using any compass setting, put the compass A parallelogram with right angles is a rectangle.

at P and draw an arc above it that intersects line a. Thus, PRSQ is a rectangle. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw .

38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Since and , the measure of Sample answer: If the diagonals of a parallelogram angles P and Q is 90 degrees. Lines that are are congruent, then it is a rectangle. He should perpendicular to the same line are parallel, so measure the diagonals of the end zone and both . The same compass setting was used to pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a locate points R and S, so . If one pair of parallelogram. If the diagonals are congruent, then opposite sides of a quadrilateral is both parallel and the end zone is a rectangle. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. ANSWER: Thus, PRSQ is a rectangle. Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are ANSWER: congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle. Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so . The same compass setting was used to

locate points R and S, so . If one pair of 39. If XW = 3, WZ = 4, and XZ = b, find YW. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. SOLUTION: A parallelogram with right angles is a rectangle. The diagonals of a rectangle are congruent to each Thus, PRSQ is a rectangle. other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 WZ .

Therefore, YW = 5.

ANSWER: 5

38. SPORTS The end zone of a football field is 160 feet 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. wide and 30 feet long. Kyle is responsible for SOLUTION: painting the field. He has finished the end zone. The diagonals of a rectangle are congruent to each Explain how Kyle can confirm that the end zone is other. So, WY = XZ = 2c. All four angles of a the regulation size and be sure that it is also a rectangle are right angles. So, is a right rectangle using only a tape measure. triangle. By the Pythagorean Theorem, XZ2 = XY2 + SOLUTION: 2 ZY . Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then the end zone is a rectangle. Therefore, WY = 2(5) = 10.

ANSWER: ANSWER: Sample answer: He should measure the diagonals of 10 the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are 41. SIGNS The sign below is in the foyer of Nyoko’s congruent, then the end zone is a rectangle. school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your ALGEBRA Quadrilateral WXYZ is a rectangle. reasoning.

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a SOLUTION: rectangle are right angles. So, is a right 2 2 triangle. By the Pythagorean Theorem, XZ = XW + Both pairs of opposite sides are congruent, so the 2 WZ . sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we Therefore, YW = 5. could prove that the sign was a rectangle.

ANSWER: ANSWER: 5 No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. rectangle. SOLUTION: PROOF Write a coordinate proof of each The diagonals of a rectangle are congruent to each statement. other. So, WY = XZ = 2c. All four angles of a 42. The diagonals of a rectangle are congruent. rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XY2 + SOLUTION: 2 Begin by positioning quadrilateral ABCD on a ZY . coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a Therefore, WY = 2(5) = 10. rectangle and you need to prove . Use the properties that you have learned about rectangles to ANSWER: walk through the proof. 10 Given: ABCD is a rectangle. 41. SIGNS The sign below is in the foyer of Nyoko’s Prove: school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your reasoning.

Proof: Use the Distance Formula to find the lengths of the diagonals. SOLUTION:

Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the The diagonals, have the same length, so diagonals could be proven to be congruent, or if the they are congruent. angles could be proven to be right angles, then we ANSWER: could prove that the sign was a rectangle. Given: ABCD is a rectangle. ANSWER: Prove: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent.

SOLUTION: Proof: Use the Distance Formula to find Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let and . the length of the bases be a units and the height be b have the same length, so they are units. Then the rest of the vertices are A(0, b), B(a, congruent. b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a 43. If the diagonals of a parallelogram are congruent, then it is a rectangle. rectangle and you need to prove . Use the properties that you have learned about rectangles to SOLUTION: walk through the proof. Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let Given: ABCD is a rectangle. the length of the bases be a units and the height be c Prove: units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Proof: Use the Distance Formula to find the lengths of the Given: ABCD and diagonals. Prove: ABCD is a rectangle.

The diagonals, have the same length, so they are congruent.

ANSWER: Proof:

Given: ABCD is a rectangle. Prove:

AC = BD. So,

Proof: Use the Distance Formula to find and . Because A and B are different points, have the same length, so they are a ≠ 0. Then b = 0. The slope of is undefined congruent. and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle. 43. If the diagonals of a parallelogram are congruent, then it is a rectangle. ANSWER:

SOLUTION: Given: ABCD and Begin by positioning parallelogram ABCD on a Prove: ABCD is a rectangle. coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given

ABCD and and you need to prove that Proof: ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and But AC = BD. So, Prove: ABCD is a rectangle.

Proof: Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

AC = BD. So, 44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the Because A and B are different points, appropriate angles and complete the table below. a ≠ 0. Then b = 0. The slope of is undefined

and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle. c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. ANSWER: Given: ABCD and SOLUTION: Prove: ABCD is a rectangle. a. Sample answer:

Proof:

But AC = BD. So, b. Use a protractor to measure each angle listed in the table.

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a Because A and B are different points, parallelogram with four congruent sides are perpendicular. a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is ANSWER: a right angle and ABCD is a rectangle.. a. Sample answer:

b. c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides.

SOLUTION: c. Sample answer: The diagonals of a parallelogram a. Sample answer: with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

b. Use a protractor to measure each angle listed in the table. SOLUTION: All four angles of a rectangle are right angles. So,

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a The diagonals of a rectangle are congruent and parallelogram with four congruent sides are bisect each other. So, is an isosceles triangle perpendicular. with ANSWER: So, a. Sample answer:

The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

ANSWER: b. x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two c. Sample answer: The diagonals of a parallelogram congruent acute triangles can be arranged to make a with four congruent sides are perpendicular. rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is 45. CHALLENGE In rectangle ABCD, either of them correct? Explain your reasoning. Find the SOLUTION: values of x and y. When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right SOLUTION: angle. All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So, Tamika is correct. So, ANSWER: Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. The sum of the angles of a triangle is 180. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right So, angle. By the Vertical Angle Theorem, 47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines?

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: Let the lines n, p, q, and r intersect the line l at the SOLUTION: points A, B, C, and D respectively. Similarly, let the When two congruent triangles are arranged to form a lines intersect the line m at the points W, X, Y, and Z. quadrilateral, two of the angles are formed by a single vertex of a triangle.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6 So, Tamika is correct. 48. OPEN ENDED Write the equations of four lines ANSWER: having intersections that form the vertices of a Tamika; Sample answer: When two congruent rectangle. Verify your answer using coordinate triangles are arranged to form a quadrilateral, two of geometry. the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of SOLUTION: the angles in the congruent triangles has to be a right Sample answer: angle. A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that 47. REASONING In the diagram at the right, lines n, p, run parallel to the x-axis will be parallel to each other q,and r are parallel and lines l and m are parallel. and perpendicular to lines parallel to the y-axis. How many rectangles are formed by the intersecting Selecting y = 0 and x = 0 will form 2 sides of the lines? rectangle. The lines y = 4 and x = 6 form the other 2 sides.

x = 0, x = 6, y = 0, y = 4;

SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z. The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 Sample answer: x = 0, x = 6, y = 0, rectangles are formed by the intersecting lines. y = 4;

ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: The length of is 6 − 0 or 6 units and the length of A rectangle has 4 right angles and each pair of is 6 – 0 or 6 units. The slope of AB is 0 and the opposite sides is parallel and congruent. Lines that slope of DC is 0. Since a pair of sides of the run parallel to the x-axis will be parallel to each other quadrilateral is both parallel and congruent, by and perpendicular to lines parallel to the y-axis. Theorem 6.12, the quadrilateral is a parallelogram. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 Since is horizontal and is vertical, the lines sides. are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by x = 0, x = 6, y = 0, y = 4; definition, the parallelogram is a rectangle.

49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not rectangles. SOLUTION: Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always The length of is 6 − 0 or 6 units and the length of parallelograms. is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the Parallelograms with right angles are rectangles, so quadrilateral is both parallel and congruent, by some parallelograms are rectangles, but others with Theorem 6.12, the quadrilateral is a parallelogram. non-right angles are not. Since is horizontal and is vertical, the lines ANSWER: are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has Sample answer: All rectangles are parallelograms one right angle, it has four right angles. Therefore, by because, by definition, both pairs of opposite sides definition, the parallelogram is a rectangle. are parallel. Parallelograms with right angles are rectangles, so some parallelograms are rectangles, ANSWER: but others with non-right angles are not. Sample answer: x = 0, x = 6, y = 0, 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM =

y = 4; 13, what values of x and y make parallelogram FGHJ a rectangle?

A x = 3, y = 4 B x = 4, y = 3 C x = 7, y = 8 The length of is 6 − 0 or 6 units and the length of D x = 8, y = 7 is 6 – 0 or 6 units. The slope of AB is 0 and the SOLUTION: slope of DC is 0. Since a pair of sides of the The opposite sides of a parallelogram are congruent. quadrilateral is both parallel and congruent, by So, FJ = GH. Theorem 6.12, the quadrilateral is a parallelogram. –3x + 5y = 11 FJ = GH Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they If the diagonals of a parallelogram are congruent, form is 90. By Theorem 6.6, if a parallelogram has then it is a rectangle. Since FGHJ is a parallelogram, one right angle, it has four right angles. Therefore, by the diagonals bisect each other. So, if it is a definition, the parallelogram is a rectangle. rectangle, then will be an isosceles triangle with That is, 3x + y = 13. WRITING IN MATH 49. Explain why all rectangles are parallelograms, but all parallelograms are not Add the two equations to eliminate the x-term and rectangles. solve for y. SOLUTION: –3x + 5y = 11 Sample answer: A parallelogram is a quadrilateral in 3x + y = 13 which each pair of opposite sides are parallel. By 6y = 24 Add to eliminate x-term. definition of rectangle, all rectangles have both pairs y = 4 Divide each side by 6. of opposite sides parallel. So rectangles are always parallelograms. Use the value of y to find the value of x. 3x + 4 = 13 Substitute. Parallelograms with right angles are rectangles, so 3x = 9 Subtract 4 from each side. some parallelograms are rectangles, but others with x = 3 Divide each side by 3. non-right angles are not. So, x = 3, y = 4, the correct choice is A. ANSWER: Sample answer: All rectangles are parallelograms ANSWER: because, by definition, both pairs of opposite sides A are parallel. Parallelograms with right angles are rectangles, so some parallelograms are rectangles, 51. ALGEBRA A rectangular playground is surrounded but others with non-right angles are not. by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = equations could be used to find r, the shorter side of 13, what values of x and y make parallelogram the playground? FGHJ a rectangle? F 10r + r = 80 G 4r + 10 = 80 H r(r + 10) = 80 J 2(r + 10) + 2r = 80 SOLUTION: A x = 3, y = 4 The formula to find the perimeter of a rectangle is B x = 4, y = 3 given by P = 2(l + w). The perimeter is given to be 8 C x = 7, y = 8 ft. The width is r and the length is r + 10. So, 2r + 2 D x = 8, y = 7 (r + 10) = 80. Therefore, the correct choice is J. SOLUTION: The opposite sides of a parallelogram are congruent. ANSWER: So, FJ = GH. J –3x + 5y = 11 FJ = GH 52. SHORT RESPONSE What is the measure of If the diagonals of a parallelogram are congruent, ? then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a rectangle, then will be an isosceles triangle with That is, 3x + y = 13.

Add the two equations to eliminate the x-term and solve for y. –3x + 5y = 11 SOLUTION: 3x + y = 13 The diagonals of a rectangle are congruent and 6y = 24 Add to eliminate x-term. bisect each other. So, is an isosceles triangle y = 4 Divide each side by 6. with So, Use the value of y to find the value of x. By the Exterior Angle Theorem, 3x + 4 = 13 Substitute. 3x = 9 Subtract 4 from each side. x = 3 Divide each side by 3. ANSWER: 112 So, x = 3, y = 4, the correct choice is A. 53. SAT/ACT If p is odd, which of the following must ANSWER: also be odd? A A 2p B 2p + 2 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 C feet longer than the other. Which of the following equations could be used to find r, the shorter side of D the playground? E p + 2 F 10r + r = 80 SOLUTION: G 4r + 10 = 80 Analyze each answer choice to determine which H r(r + 10) = 80 expression is odd when p is odd. J 2(r + 10) + 2r = 80 Choice A 2p This is a multiple of 2 so it is even. SOLUTION: Choice B 2p + 2 The formula to find the perimeter of a rectangle is This can be rewritten as 2(p + 1) which is also a given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 multiple of 2 so it is even.

(r + 10) = 80. Choice C Therefore, the correct choice is J. If p is odd , is not a whole number. ANSWER: Choice D 2p – 2 J This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. 52. SHORT RESPONSE What is the measure of Choice E p + 2 ? Adding 2 to an odd number results in an odd number. This is odd. The correct choice is E.

ANSWER: E

SOLUTION: ALGEBRA Find x and y so that the The diagonals of a rectangle are congruent and quadrilateral is a parallelogram. bisect each other. So, is an isosceles triangle with So, By the Exterior Angle Theorem, 54. SOLUTION: ANSWER: The opposite sides of a parallelogram are congruent. 112 53. SAT/ACT If p is odd, which of the following must also be odd? A 2p In a parallelogram, consecutive interior angles are B 2p + 2 supplementary.

C D E p + 2 SOLUTION: Analyze each answer choice to determine which ANSWER: expression is odd when p is odd. x = 2, y = 41 Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

Choice C 55. If p is odd , is not a whole number. SOLUTION: Choice D 2p – 2 The opposite sides of a parallelogram are congruent. This can be rewritten as 2( p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd The opposite sides of a parallelogram are parallel to number. This is odd. each other. So, the angles with the measures (3y +

The correct choice is E. 3)° and (4y – 19)° are alternate interior angles and ANSWER: hence they are equal.

ALGEBRA Find x and y so that the quadrilateral is a parallelogram. ANSWER: x = 8, y = 22

54. SOLUTION: The opposite sides of a parallelogram are congruent. 56.

SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. In a parallelogram, consecutive interior angles are Solve the first equation to find the value of y. supplementary. 2y = 14

y = 7 Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x 2 = x ANSWER: x = 2, y = 41 ANSWER: x = 2, y = 7

57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). 55. SOLUTION: The diagonals of a parallelogram bisect each other, SOLUTION: so the intersection is the midpoint of either diagonal. The opposite sides of a parallelogram are congruent. Find the midpoint of or .

The opposite sides of a parallelogram are parallel to Therefore, the coordinates of the intersection of the each other. So, the angles with the measures (3y + diagonals are (2.5, 0.5). 3)° and (4y – 19)° are alternate interior angles and hence they are equal. ANSWER: (2.5, 0.5)

Refer to the figure.

ANSWER: x = 8, y = 22

56. 58. If , name two congruent angles. SOLUTION: SOLUTION: The diagonals of a parallelogram bisect each other. Since is an isosceles triangle. So, So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 ANSWER: y = 7 Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 59. If , name two congruent segments. 4 = 2x SOLUTION: 2 = x In a triangle with two congruent angles, the sides ANSWER: opposite to the congruent angles are also congruent. x = 2, y = 7 So, .

57. COORDINATE GEOMETRY Find the ANSWER: coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). 60. If , name two congruent segments. SOLUTION: SOLUTION: The diagonals of a parallelogram bisect each other, In a triangle with two congruent angles, the sides so the intersection is the midpoint of either diagonal. opposite to the congruent angles are also congruent. Find the midpoint of or . So, .

ANSWER:

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem, FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps Therefore, to prevent the door from warping over time.

If feet, PS = 7 feet, and ANSWER: , find each measure. 33.5 4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore,

1. QR ANSWER: SOLUTION: 56.5 The opposite sides of a rectangle are parallel and ALGEBRA Quadrilateral DEFG is a rectangle. congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: SOLUTION: The diagonals of a rectangle bisect each other. So, The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: ANSWER: 11

6. If , find . 3. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Therefore, ANSWER: ANSWER: 51 33.5 7. PROOF If ABDE is a rectangle and , 4. prove that . SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore, SOLUTION: ANSWER: You need to walk through the proof step by step. 56.5 Look over what you are given and what you need to prove. You are given ABDE is a rectangle and ALGEBRA Quadrilateral DEFG is a rectangle. . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABDE is a rectangle;

5. If FD = 3x – 7 and EG = x + 5, find EG. Prove: SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. Statements(Reasons) EG = 6 + 5 = 11 1. ABDE is a rectangle; . (Given) ANSWER: 2. ABDE is a parallelogram. (Def. of 6-4 Rectangles11 rectangle) 3. (Opp. sides of a .) 6. If , 4. are right angles.(Def. of rectangle) find . 5. (All rt .)

SOLUTION: 6. (SAS) All four angles of a rectangle are right angles. So, 7. (CPCTC)

ANSWER:

Given: ABDE is a rectangle; Prove:

ANSWER: 51

7. PROOF If ABDE is a rectangle and , Statements(Reasons)

prove that . 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. Look over what you are given and what you need to COORDINATE GEOMETRY Graph each prove. You are given ABDE is a rectangle and quadrilateral with the given vertices. Determine . You need to prove . Use the whether the figure is a rectangle. Justify your properties that you have learned about rectangles to answer using the indicated formula. walk through the proof. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION: Given: ABDE is a rectangle; Prove:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) Use the slope formula to find the slope of the sides of 2. ABDE is a parallelogram. (Def. of the quadrilateral. rectangle) 3. (Opp. sides of a .) eSolutions4. Manual - Powered are right angles.(Def. of rectangle)by Cognero Page 2 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; There are no parallel sides, so WXYZ is not a Prove: parallelogram. Therefore, WXYZ is not a rectangle. ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of

slope of , and slope of slope of Statements(Reasons) , so WXYZ is not a parallelogram. Therefore, 1. ABDE is a rectangle; . WXYZ is not a rectangle. (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula Formula SOLUTION: SOLUTION:

Use the slope formula to find the slope of the sides of First, Use the Distance formula to find the lengths of the quadrilateral. the sides of the quadrilateral.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals. There are no parallel sides, so WXYZ is not a

parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER: So the diagonals are congruent. Thus, ABCD is a rectangle. No; slope of , slope of , slope of ANSWER: , and slope of . Slope of Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are slope of , and slope of slope of congruent. Thus, ABCD is a rectangle. , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance AD = 2 feet, and , find each Formula measure. SOLUTION:

10. BC SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. First, Use the Distance formula to find the lengths of the sides of the quadrilateral. ANSWER: 2 ft

11. DB

so ABCD is a SOLUTION: parallelogram. All four angles of a rectangle are right angles. So, A parallelogram is a rectangle if the diagonals are is a right triangle. By the Pythagorean 2 2 2 congruent. Use the Distance formula to find the Theorem, DB = BC + DC .

lengths of the diagonals. BC = AD = 2

DC = AB = 6

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a 6.3 ft parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. 12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

By the Vertical Angle Theorem,

ANSWER: FENCING X-braces are also used to provide 50 support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each 13. measure. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, All four angles of a rectangle are right angles.

10. BC ANSWER: SOLUTION: 25 The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. CCSS REGULARITY Quadrilateral WXYZ is a rectangle. ANSWER: 2 ft

11. DB

SOLUTION: All four angles of a rectangle are right angles. So, 14. If ZY = 2x + 3 and WX = x + 4, find WX. is a right triangle. By the Pythagorean 2 2 2 SOLUTION: Theorem, DB = BC + DC . The opposite sides of a rectangle are parallel and BC = AD = 2 congruent. So, ZY = WX. DC = AB = 6 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5.

ANSWER: ANSWER: 6.3 ft 5 12. 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and The diagonals of a rectangle bisect each other. So, bisect each other. So, is an isosceles PY = WP. triangle.

Then,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

By the Vertical Angle Theorem,

ANSWER: 50 ANSWER: 43 13. 16. If , find SOLUTION: . The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles SOLUTION: triangle. All four angles of a rectangle are right angles. So, Then, All four angles of a rectangle are right angles.

ANSWER: 25 ANSWER: CCSS REGULARITY Quadrilateral WXYZ is a 39 rectangle. 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY.

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: Then ZP = 4(7) – 9 = 19. The opposite sides of a rectangle are parallel and Therefore, ZX = 2(ZP) = 38. congruent. So, ZY = WX. 2x + 3 = x + 4 ANSWER: x = 1 38 Use the value of x to find WX. WX = 1 + 4 = 5. 18. If , find . ANSWER: 5 SOLUTION: All four angles of a rectangle are right angles. So, 15. If PY = 3x – 5 and WP = 2x + 11, find ZP.

SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP.

The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 48

19. If , find . SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 43 is a right triangle. Then,

16. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Therefore, Since is a

transversal of parallel sides and , alternate

interior angles WZX and ZXY are congruent. So, = = 46.

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to Then ZP = 4(7) – 9 = 19. prove. Here, you are given ABCD is a rectangle. Therefore, ZX = 2(ZP) = 38. You need to prove . Use the properties that you have learned about rectangles to ANSWER: walk through the proof. 38 Given: ABCD is a rectangle. 18. If , find Prove: . Proof: SOLUTION: Statements (Reasons)

All four angles of a rectangle are right angles. So, 1. ABCD is a rectangle. (Given)

2. ABCD is a parallelogram. (Def. of rectangle)

3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS) ANSWER: ANSWER: Proof: 48 Statements (Reasons) 1. ABCD is a rectangle. (Given) 19. If , find 2. ABCD is a parallelogram. (Def. of rectangle) . 3. (Opp. sides of a .) SOLUTION: 4. (Refl. Prop.) All four angles of a rectangle are right angles. So, 5. (Diagonals of a rectangle are .) is a right triangle. Then, 6. (SSS)

21. Given: QTVW is a rectangle.

Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46. Prove: ANSWER: SOLUTION: 46 You need to walk through the proof step by step. PROOF Write a two-column proof. Look over what you are given and what you need to 20. Given: ABCD is a rectangle. prove. Here, you are given QTVW is a rectangle and Prove: . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: QTVW is a rectangle.

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to Prove: walk through the proof. Proof: Statements (Reasons) Given: ABCD is a rectangle. 1. QTVW is a rectangle; .(Given) Prove: 2. QTVW is a parallelogram. (Def. of rectangle) Proof: 3. (Opp sides of a .) Statements (Reasons) 4. are right angles.(Def. of rectangle) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 5. (All rt .) 3. (Opp. sides of a .) 6. QR = ST (Def. of segs.)

4. (Refl. Prop.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 5. (Diagonals of a rectangle are .) 9. QR + RS = RS + ST (Add. prop.) 6. (SSS) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) ANSWER: 11. QS = RT (Subst.) Proof: 12. (Def. of segs.) Statements (Reasons) 13. (SAS) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) ANSWER: 3. (Opp. sides of a .) Proof: Statements (Reasons) 4. (Refl. Prop.) 1. QTVW is a rectangle; .(Given) 5. (Diagonals of a rectangle are .) 2. QTVW is a parallelogram. (Def. of rectangle) 6. (SSS) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 21. Given: QTVW is a rectangle. 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) Prove: 11. QS = RT (Subst.)

SOLUTION: 12. (Def. of segs.) You need to walk through the proof step by step. 13. (SAS) Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine . You need to prove . Use whether the figure is a rectangle. Justify your the properties that you have learned about rectangles answer using the indicated formula. to walk through the proof. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula

Given: QTVW is a rectangle. SOLUTION:

Prove: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) Use the slope formula to find the slope of the sides of the quadrilateral. 4. are right angles.(Def. of rectangle)

5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS) The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. ANSWER: Therefore, the quadrilateral WXYZ is a parallelogram. Proof: The products of the slopes of the adjacent sides are Statements (Reasons) –1. So, any two adjacent sides are perpendicular to 1. QTVW is a rectangle; .(Given) each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle. 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) ANSWER: 4. are right angles.(Def. of rectangle) Yes; slope of slope of , slope 5. (All rt .) 6. QR = ST (Def. of segs.) of slope of . So WXYZ is a 7. (Refl. Prop.) parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are 8. RS = RS (Def. of segs.) perpendicular and form right angles. Thus, WXYZ is a 9. QR + RS = RS + ST (Add. prop.) rectangle. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

SOLUTION:

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. so JKLM is a Therefore, the quadrilateral WXYZ is a parallelogram. parallelogram. The products of the slopes of the adjacent sides are A parallelogram is a rectangle if the diagonals are –1. So, any two adjacent sides are perpendicular to congruent. Use the Distance formula to find the each other. That is, all the four angles are right lengths of the diagonals. angles. Therefore, WXYZ is a rectangle. ANSWER: So the diagonals are not congruent. Thus, JKLM is Yes; slope of slope of , slope not a rectangle.

of slope of . So WXYZ is a ANSWER: parallelogram. The product of the slopes of No; so JKLM is a consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a parallelogram; so rectangle. the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION: SOLUTION:

First, use the Distance Formula to find the lengths of First, Use the Distance formula to find the lengths of the sides of the quadrilateral. the sides of the quadrilateral.

so QRST is a so JKLM is a parallelogram. parallelogram. A parallelogram is a rectangle if the diagonals are A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the congruent. Use the Distance formula to find the lengths of the diagonals. lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a So the diagonals are not congruent. Thus, JKLM is rectangle. not a rectangle. ANSWER: ANSWER: Yes; so QRST is a No; so JKLM is a parallelogram. so the diagonals are parallelogram; so congruent. QRST is a rectangle. the diagonals are not congruent. Thus, JKLM is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula

SOLUTION: 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, use the slope formula to find the lengths of the First, use the Distance Formula to find the lengths of sides of the quadrilateral. the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals. The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a parallelogram. So the diagonals are congruent. Thus, QRST is a None of the adjacent sides have slopes whose rectangle. product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle. ANSWER: Yes; so QRST is a ANSWER: parallelogram. so the diagonals are No; slope of slope of and slope of congruent. QRST is a rectangle. slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

26. SOLUTION: First, use the slope formula to find the lengths of the All four angles of a rectangle are right angles. So, sides of the quadrilateral.

ANSWER: 50

27. SOLUTION: The slopes of each pair of opposite sides are equal. The measures of angles 2 and 3 are equal as they So, the two pairs of opposite sides are parallel. are alternate interior angles. Therefore, the quadrilateral GHJK is a Since the diagonals of a rectangle are congruent and parallelogram. bisect each other, the triangle with the angles 3, 7 None of the adjacent sides have slopes whose and 8 is an isosceles triangle. So, m∠7 = m∠3. product is –1. So, the angles are not right angles. Therefore, m∠7 = m∠2 = 40. Therefore, GHJK is not a rectangle. ANSWER: ANSWER: 40 No; slope of slope of and slope of 28. slope of . So, GHJK is a SOLUTION: parallelogram. The product of the slopes of The measures of angles 2 and 3 are equal as they consecutive sides ≠ –1, so the consecutive sides are are alternate interior angles. not perpendicular. Thus, GHJK is not a rectangle. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m 4 = m 1 = 50. Quadrilateral ABCD is a rectangle. Find each ∠ ∠ Since the diagonals of a rectangle are congruent and measure if . bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER:

26. 80 SOLUTION: 30. All four angles of a rectangle are right angles. So, SOLUTION:

All four angles of a rectangle are right angles. So,

ANSWER: 50 The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. 27. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m 6 = The measures of angles 2 and 3 are equal as they ∠ are alternate interior angles. m∠4 = 50. Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 3, 7 50 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. 31. ANSWER: SOLUTION: 40 The measures of angles 2 and 3 are equal as they are alternate interior angles. 28. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m 3 = m 7 = 40. The measures of angles 2 and 3 are equal as they ∠ ∠ The sum of the three angles of a triangle is 180. are alternate interior angles. Therefore, m∠8 = 180 – (40 + 40) = 100. Therefore, m∠3 = m∠2 = 40. ANSWER: ANSWER: 100 40 32. CCSS MODELING Jody is building a new 29. bookshelf using wood and metal supports like the one SOLUTION: shown. To what length should she cut the metal All four angles of a rectangle are right angles. So, supports in order for the bookshelf to be square, which means that the angles formed by the shelves and the vertical supports are all right angles? Explain

your reasoning. The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180.

Therefore, m∠5 = 180 – (50 + 50) = 80. SOLUTION: ANSWER: In order for the bookshelf to be square, the lengths of 80 all of the metal supports should be equal. Since we know the length of the bookshelves and the distance 30. between them, use the Pythagorean Theorem to find SOLUTION: the lengths of the metal supports. Let x be the length of each support. Express 3 feet All four angles of a rectangle are right angles. So, as 36 inches.

SOLUTION: PROOF Write a two-column proof. The measures of angles 2 and 3 are equal as they 33. Theorem 6.13 are alternate interior angles. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 You need to walk through the proof step by step. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. Look over what you are given and what you need to The sum of the three angles of a triangle is 180. prove. Here, you are given WXYZ is a rectangle with Therefore, m∠8 = 180 – (40 + 40) = 100. diagonals . You need to prove ANSWER: . Use the properties that you have learned about rectangles to walk through the proof. 100

32. CCSS MODELING Jody is building a new Given: WXYZ is a rectangle with diagonals bookshelf using wood and metal supports like the one . shown. To what length should she cut the metal Prove: supports in order for the bookshelf to be square, which means that the angles formed by the shelves and the vertical supports are all right angles? Explain your reasoning.

Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of SOLUTION: rectangle) In order for the bookshelf to be square, the lengths of 5. (All right .) all of the metal supports should be equal. Since we 6. (SAS) know the length of the bookshelves and the distance 7. (CPCTC) between them, use the Pythagorean Theorem to find

the lengths of the metal supports. ANSWER: Let x be the length of each support. Express 3 feet as 36 inches. Given: WXYZ is a rectangle with diagonals . Prove:

Therefore, the metal supports should each be 39 inches long or 3 feet 3 inches long. Proof: ANSWER: 1. WXYZ is a rectangle with diagonals . 3 ft 3 in.; Sample answer: In order for the bookshelf

to be square, the lengths of all of the metal supports (Given) should be equal. Since I knew the length of the 2. (Opp. sides of a .) bookshelves and the distance between them, I used 3. (Refl. Prop.) the Pythagorean Theorem to find the lengths of the 4. are right angles. (Def. of metal supports. rectangle) PROOF Write a two-column proof. 5. (All right .) 33. Theorem 6.13 6. (SAS) SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. Look over what you are given and what you need to 34. Theorem 6.14 prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove SOLUTION: You need to walk through the proof step by step. . Use the properties that you have learned Look over what you are given and what you need to about rectangles to walk through the proof. prove. Here, you are given

. You need to Given: WXYZ is a rectangle with diagonals prove that WXYZ is a rectangle.. Use the properties . that you have learned about rectangles to walk Prove: through the proof.

Given: Prove: WXYZ is a rectangle.

Proof: 1. WXYZ is a rectangle with diagonals . (Given)

2. (Opp. sides of a .) Proof: 3. (Refl. Prop.) 1. (Given) 4. are right angles. (Def. of 2. (SSS) rectangle) 3. (CPCTC) 5. (All right .) 4. (Def. of ) 6. (SAS) 5. WXYZ is a parallelogram. (If both pairs of opp. 7. (CPCTC) sides are , then quad. is .) 6. are supplementary. (Cons. ANSWER: are suppl.) Given: WXYZ is a rectangle with diagonals 7. (Def of suppl.)

. 8. are right angles. (If Prove: 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

6. (SAS) 1. (Given) 2. (SSS) 7. (CPCTC) 3. (CPCTC) 4. (Def. of ) 34. Theorem 6.14 5. WXYZ is a parallelogram. (If both pairs of opp. SOLUTION: sides are , then quad. is .) You need to walk through the proof step by step. 6. are supplementary. (Cons. Look over what you are given and what you need to are suppl.) prove. Here, you are given 7. (Def of suppl.) . You need to 8. are right angles. (If prove that WXYZ is a rectangle.. Use the properties

that you have learned about rectangles to walk 2 and suppl., each is a rt. .) through the proof. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) Given: 10. WXYZ is a rectangle. (Def. of rectangle) Prove: WXYZ is a rectangle. PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION:

sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) ABCD is a parallelogram, and is a right angle. 9. are right angles. (If a has 1 Since ABCD is a parallelogram and has one right rt. , it has 4 rt. .) angle, then it has four right angles. So by the 10. WXYZ is a rectangle. (Def. of rectangle) definition of a rectangle, ABCD is a rectangle.

ANSWER: ANSWER: Given: Prove: WXYZ is a rectangle.

ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right Proof: angle, then it has four right angles. So by the 1. (Given) definition of a rectangle, ABCD is a rectangle. 2. (SSS) 3. (CPCTC) 36. If a quadrilateral has four right angles, then it is a 4. (Def. of ) rectangle. 5. WXYZ is a parallelogram. (If both pairs of opp. SOLUTION: sides are , then quad. is .) You need to walk through the proof step by step. 6. are supplementary. (Cons. Look over what you are given and what you need to are suppl.) prove. Here, you are given a quadrilateral with four 7. (Def of suppl.) right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. 8. are right angles. (If Then use the properties of parallelograms and 2 and suppl., each is a rt. .) rectangles to walk through the proof. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of SOLUTION: opposite angles are congruent. By definition of a You need to walk through the proof step by step. rectangle, ABCD is a rectangle. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one ANSWER: right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

ABCD is a parallelogram, and is a right angle. 37. CONSTRUCTION Construct a rectangle using the Since ABCD is a parallelogram and has one right construction for congruent segments and the angle, then it has four right angles. So by the construction for a line perpendicular to another line definition of a rectangle, ABCD is a rectangle. through a point on the line. Justify each step of the construction. ANSWER: SOLUTION: Sample answer: Step 1: Graph a line and place points P and Q on the line.

definition of a rectangle, ABCD is a rectangle. Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. 36. If a quadrilateral has four right angles, then it is a Place the compass on each point of intersection and rectangle. draw arcs that intersect above . SOLUTION: You need to walk through the proof step by step. Step 4: Draw line a through point P and the Look over what you are given and what you need to intersection of the arcs drawn in step 3. prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Step 5: Repeat steps 2 and 3 using point Q. Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and Step 6: Draw line b through point Q and the rectangles to walk through the proof. intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the ABCD is a quadrilateral with four right angles. compass at Q and draw an arc above it that ABCD is a parallelogram because both pairs of intersects line b. Label the point of intersection S. opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 9: Draw . ANSWER:

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Since and , the measure of 37. CONSTRUCTION Construct a rectangle using the angles P and Q is 90 degrees. Lines that are construction for congruent segments and the perpendicular to the same line are parallel, so construction for a line perpendicular to another line . The same compass setting was used to through a point on the line. Justify each step of the construction. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and SOLUTION: congruent, then the quadrilateral is a parallelogram. Sample answer: A parallelogram with right angles is a rectangle. Step 1: Graph a line and place points P and Q on the Thus, PRSQ is a rectangle. line.

draw arcs that intersect above . opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 4: Draw line a through point P and the A parallelogram with right angles is a rectangle. intersection of the arcs drawn in step 3. Thus, PRSQ is a rectangle.

Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

the end zone is a rectangle. Since and , the measure of ANSWER: angles P and Q is 90 degrees. Lines that are Sample answer: He should measure the diagonals of perpendicular to the same line are parallel, so the end zone and both pairs of opposite sides. If the . The same compass setting was used to diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and ALGEBRA Quadrilateral WXYZ is a rectangle. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

ANSWER: 39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: Sample answer: Since and , the The diagonals of a rectangle are congruent to each measure of angles P and Q is 90 degrees. Lines that other. So, YW = XZ = b. All four angles of a are perpendicular to the same line are parallel, so rectangle are right angles. So, is a right 2 2 . The same compass setting was used to triangle. By the Pythagorean Theorem, XZ = XW + 2 locate points R and S, so . If one pair of WZ . opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Therefore, YW = 5. ANSWER: 5

SOLUTION: Therefore, WY = 2(5) = 10. Sample answer: If the diagonals of a parallelogram ANSWER: are congruent, then it is a rectangle. He should 10 measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides 41. SIGNS The sign below is in the foyer of Nyoko’s are congruent, then the quadrilateral is a school. Based on the dimensions given, can Nyoko parallelogram. If the diagonals are congruent, then be sure that the sign is a rectangle? Explain your the end zone is a rectangle. reasoning. ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

SOLUTION:

Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that 39. If XW = 3, WZ = 4, and XZ = b, find YW. can be used to prove that it is a rectangle. If the SOLUTION: diagonals could be proven to be congruent, or if the The diagonals of a rectangle are congruent to each angles could be proven to be right angles, then we other. So, YW = XZ = b. All four angles of a could prove that the sign was a rectangle. rectangle are right angles. So, is a right ANSWER: 2 2 triangle. By the Pythagorean Theorem, XZ = XW + No; sample answer: Both pairs of opposite sides are 2 WZ . congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

Therefore, YW = 5. PROOF Write a coordinate proof of each statement. ANSWER: 42. The diagonals of a rectangle are congruent. 5 SOLUTION: Begin by positioning quadrilateral ABCD on a 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b SOLUTION: units. Then the rest of the vertices are A(0, b), B(a, The diagonals of a rectangle are congruent to each b), and C(a, 0). You need to walk through the proof other. So, WY = XZ = 2c. All four angles of a step by step. Look over what you are given and what rectangle are right angles. So, is a right you need to prove. Here, you are given ABCD is a 2 2 triangle. By the Pythagorean Theorem, XZ = XY + rectangle and you need to prove . Use the 2 ZY . properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove:

Therefore, WY = 2(5) = 10.

ANSWER: 10

41. SIGNS The sign below is in the foyer of Nyoko’s

school. Based on the dimensions given, can Nyoko Proof: be sure that the sign is a rectangle? Explain your Use the Distance Formula to find the lengths of the reasoning. diagonals.

The diagonals, have the same length, so they are congruent.

ANSWER: SOLUTION: Given: ABCD is a rectangle. Prove: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are Proof: congruent, so the sign is a parallelogram, but no Use the Distance Formula to find measure is given that can be used to prove that it is a and . rectangle. have the same length, so they are PROOF Write a coordinate proof of each congruent. statement. 42. The diagonals of a rectangle are congruent. 43. If the diagonals of a parallelogram are congruent, SOLUTION: then it is a rectangle. Begin by positioning quadrilateral ABCD on a SOLUTION: coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b Begin by positioning parallelogram ABCD on a units. Then the rest of the vertices are A(0, b), B(a, coordinate plane. Place vertex A at the origin. Let b), and C(a, 0). You need to walk through the proof the length of the bases be a units and the height be c step by step. Look over what you are given and what units. Then the rest of the vertices are B(a, 0), C(b + you need to prove. Here, you are given ABCD is a a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given rectangle and you need to prove . Use the and what you need to prove. Here, you are given properties that you have learned about rectangles to walk through the proof. ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you Given: ABCD is a rectangle. have learned about rectangles to walk through the proof. Prove: Given: ABCD and Prove: ABCD is a rectangle.

Proof: Use the Distance Formula to find the lengths of the diagonals. Proof:

The diagonals, have the same length, so AC = BD. So, they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.

Proof: ANSWER: Use the Distance Formula to find Given: ABCD and and . Prove: ABCD is a rectangle. have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION:

a, c), and D(b, c). You need to walk through the But AC = BD. So, proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Prove: ABCD is a rectangle. Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special

Proof: parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and

label the intersections R. AC = BD. So, b. TABULAR Use a protractor to measure the

appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: Because A and B are different points, a. Sample answer: a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.

ANSWER: Given: ABCD and Prove: ABCD is a rectangle.

b. Use a protractor to measure each angle listed in the table. Proof:

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each But AC = BD. So, formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular. ANSWER: a. Sample answer:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one b. parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the c. Sample answer: The diagonals of a parallelogram appropriate angles and complete the table below. with four congruent sides are perpendicular. 45. CHALLENGE In rectangle ABCD, Find the c. VERBAL Make a conjecture about the diagonals values of x and y. of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and b. Use a protractor to measure each angle listed in bisect each other. So, is an isosceles triangle the table. with So,

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are

perpendicular. The sum of the angles of a triangle is 180. So, ANSWER: By the Vertical Angle Theorem, a. Sample answer:

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. b. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a c. Sample answer: The diagonals of a parallelogram single vertex of a triangle. with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

So, angle.

47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines? The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

SOLUTION: ANSWER: Let the lines n, p, q, and r intersect the line l at the x = 6, y = −10 points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z. 46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

In order for the quadrilateral to be a rectangle, one of ANSWER: the angles in the congruent triangles has to be a right 6 angle. 48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: So, Tamika is correct. A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that ANSWER: run parallel to the x-axis will be parallel to each other Tamika; Sample answer: When two congruent and perpendicular to lines parallel to the y-axis. triangles are arranged to form a quadrilateral, two of Selecting y = 0 and x = 0 will form 2 sides of the the angles are formed by a single vertex of a triangle. rectangle. The lines y = 4 and x = 6 form the other 2 In order for the quadrilateral to be a rectangle, one of sides. the angles in the congruent triangles has to be a right angle. x = 0, x = 6, y = 0, y = 4;

47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines?

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by SOLUTION: Theorem 6.12, the quadrilateral is a parallelogram. Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the Since is horizontal and is vertical, the lines lines intersect the line m at the points W, X, Y, and Z. are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6 The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the 48. OPEN ENDED Write the equations of four lines slope of DC is 0. Since a pair of sides of the having intersections that form the vertices of a quadrilateral is both parallel and congruent, by rectangle. Verify your answer using coordinate Theorem 6.12, the quadrilateral is a parallelogram. geometry. Since is horizontal and is vertical, the lines SOLUTION: are perpendicular and the measure of the angle they Sample answer: form is 90. By Theorem 6.6, if a parallelogram has A rectangle has 4 right angles and each pair of one right angle, it has four right angles. Therefore, by opposite sides is parallel and congruent. Lines that definition, the parallelogram is a rectangle. run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. 49. WRITING IN MATH Explain why all rectangles Selecting y = 0 and x = 0 will form 2 sides of the are parallelograms, but all parallelograms are not rectangle. The lines y = 4 and x = 6 form the other 2 rectangles. sides. SOLUTION:

Sample answer: A parallelogram is a quadrilateral in x = 0, x = 6, y = 0, y = 4; which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms.

Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

ANSWER:

Sample answer: All rectangles are parallelograms The length of is 6 − 0 or 6 units and the length of because, by definition, both pairs of opposite sides is 6 – 0 or 6 units. The slope of AB is 0 and the are parallel. Parallelograms with right angles are slope of DC is 0. Since a pair of sides of the rectangles, so some parallelograms are rectangles, quadrilateral is both parallel and congruent, by but others with non-right angles are not. Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = are perpendicular and the measure of the angle they 13, what values of x and y make parallelogram

form is 90. By Theorem 6.6, if a parallelogram has FGHJ a rectangle? one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, A y = 4; x = 3, y = 4 B x = 4, y = 3 C x = 7, y = 8 D x = 8, y = 7 SOLUTION: The opposite sides of a parallelogram are congruent. So, FJ = GH. –3x + 5y = 11 FJ = GH

If the diagonals of a parallelogram are congruent, The length of is 6 − 0 or 6 units and the length of then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the rectangle, then will be an isosceles triangle quadrilateral is both parallel and congruent, by with That is, 3x + y = 13. Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines Add the two equations to eliminate the x-term and

are perpendicular and the measure of the angle they solve for y.

form is 90. By Theorem 6.6, if a parallelogram has –3x + 5y = 11

one right angle, it has four right angles. Therefore, by 3x + y = 13

definition, the parallelogram is a rectangle. 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6. 49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not Use the value of y to find the value of x. rectangles. 3x + 4 = 13 Substitute. 3x = 9 Subtract 4 from each side. SOLUTION: x = 3 Divide each side by 3. Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By So, x = 3, y = 4, the correct choice is A. definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always ANSWER: parallelograms. A

Parallelograms with right angles are rectangles, so 51. ALGEBRA A rectangular playground is surrounded some parallelograms are rectangles, but others with by an 80-foot fence. One side of the playground is 10 non-right angles are not. feet longer than the other. Which of the following equations could be used to find r, the shorter side of ANSWER: the playground? Sample answer: All rectangles are parallelograms F 10r + r = 80 because, by definition, both pairs of opposite sides G 4r + 10 = 80 are parallel. Parallelograms with right angles are H r(r + 10) = 80 rectangles, so some parallelograms are rectangles, J 2(r + 10) + 2r = 80 but others with non-right angles are not. SOLUTION: 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = The formula to find the perimeter of a rectangle is 13, what values of x and y make parallelogram given by P = 2(l + w). The perimeter is given to be 8 FGHJ a rectangle? ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. Therefore, the correct choice is J.

ANSWER: J A x = 3, y = 4 B x = 4, y = 3 52. SHORT RESPONSE What is the measure of C x = 7, y = 8 ? D x = 8, y = 7 SOLUTION: The opposite sides of a parallelogram are congruent. So, FJ = GH. –3x + 5y = 11 FJ = GH

So, x = 3, y = 4, the correct choice is A. D E p + 2 ANSWER: SOLUTION: A Analyze each answer choice to determine which

51. ALGEBRA A rectangular playground is surrounded expression is odd when p is odd. by an 80-foot fence. One side of the playground is 10 Choice A 2p feet longer than the other. Which of the following This is a multiple of 2 so it is even. equations could be used to find r, the shorter side of Choice B 2p + 2 the playground? This can be rewritten as 2(p + 1) which is also a F 10r + r = 80 multiple of 2 so it is even. G 4r + 10 = 80 Choice C H r(r + 10) = 80 J 2(r + 10) + 2r = 80 If p is odd , is not a whole number. Choice D 2p – 2 SOLUTION: This can be rewritten as 2(p - 1) which is also a The formula to find the perimeter of a rectangle is multiple of 2 so it is even. given by P = 2(l + w). The perimeter is given to be 8 Choice E p + 2 ft. The width is r and the length is r + 10. So, 2r + 2 Adding 2 to an odd number results in an odd (r + 10) = 80. Therefore, the correct choice is J. number. This is odd. The correct choice is E. ANSWER: ANSWER: J E 52. SHORT RESPONSE What is the measure of ? ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

54. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and The opposite sides of a parallelogram are congruent. bisect each other. So, is an isosceles triangle with So, By the Exterior Angle Theorem, In a parallelogram, consecutive interior angles are supplementary.

ANSWER: 112

53. SAT/ACT If p is odd, which of the following must also be odd? A 2p ANSWER: B 2p + 2 x = 2, y = 41

C D E p + 2 SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. 55. Choice A 2p SOLUTION: This is a multiple of 2 so it is even. The opposite sides of a parallelogram are congruent. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

Choice C The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + If p is odd , is not a whole number. 3)° and (4y – 19)° are alternate interior angles and Choice D 2p – 2 hence they are equal. This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. ANSWER: The correct choice is E. x = 8, y = 22 ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram. 56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. 54. Solve the first equation to find the value of y. 2y = 14 SOLUTION: y = 7 The opposite sides of a parallelogram are congruent. Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x In a parallelogram, consecutive interior angles are 2 = x supplementary. ANSWER:

x = 2, y = 7

57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). ANSWER: SOLUTION: x = 2, y = 41 The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the 55. diagonals are (2.5, 0.5). SOLUTION: ANSWER: The opposite sides of a parallelogram are congruent. (2.5, 0.5) Refer to the figure.

The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

58. If , name two congruent angles. ANSWER: SOLUTION: x = 8, y = 22 Since is an isosceles triangle. So,

ANSWER:

56.

SOLUTION: 59. If , name two congruent segments. The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. SOLUTION: Solve the first equation to find the value of y. In a triangle with two congruent angles, the sides 2y = 14 opposite to the congruent angles are also congruent. y = 7 So, . Use the value of y in the second equation and solve for x. ANSWER: 7 + 3 = 2x + 6 4 = 2x 2 = x 60. If , name two congruent segments. ANSWER: SOLUTION: x = 2, y = 7 In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. 57. COORDINATE GEOMETRY Find the So, . coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C ANSWER: (4, –2), and D(–1, –1). SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem, FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. Therefore, If feet, PS = 7 feet, and ANSWER: , find each measure. 33.5

4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore,

1. QR ANSWER: SOLUTION: 56.5 The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft. ALGEBRA Quadrilateral DEFG is a rectangle.

ANSWER: 7 ft

2. SQ 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle bisect each other. So, SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: ANSWER: 11

6. If , 3. find . SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, The diagonals of a rectangle are congruent and

bisect each other. So, is an isosceles triangle.

Then, By the Exterior Angle Theorem,

Therefore,

ANSWER: ANSWER: 33.5 51

4. 7. PROOF If ABDE is a rectangle and ,

SOLUTION: prove that . The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore,

ANSWER: SOLUTION: 56.5 You need to walk through the proof step by step. Look over what you are given and what you need to ALGEBRA Quadrilateral DEFG is a rectangle. prove. You are given ABDE is a rectangle and . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABDE is a rectangle; 5. If FD = 3x – 7 and EG = x + 5, find EG. Prove: SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. Statements(Reasons) EG = 6 + 5 = 11 1. ABDE is a rectangle; . ANSWER: (Given) 2. ABDE is a parallelogram. (Def. of 11 rectangle) 3. (Opp. sides of a .) 6. If , 4. are right angles.(Def. of rectangle) find . 5. (All rt .) SOLUTION: 6. (SAS) All four angles of a rectangle are right angles. So, 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

ANSWER: 51

You need to walk through the proof step by step. 7. (CPCTC) Look over what you are given and what you need to COORDINATE GEOMETRY Graph each prove. You are given ABDE is a rectangle and quadrilateral with the given vertices. Determine . You need to prove . Use the whether the figure is a rectangle. Justify your properties that you have learned about rectangles to answer using the indicated formula. walk through the proof. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula

SOLUTION: Given: ABDE is a rectangle; Prove:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) Use the slope formula to find the slope of the sides of 2. ABDE is a parallelogram. (Def. of the quadrilateral. rectangle)

3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; There are no parallel sides, so WXYZ is not a Prove: parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of

Statements(Reasons) slope of , and slope of slope of 1. ABDE is a rectangle; . , so WXYZ is not a parallelogram. Therefore, (Given) WXYZ is not a rectangle. 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each 6-4 Rectanglesquadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION: SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral. First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the There are no parallel sides, so WXYZ is not a lengths of the diagonals. parallelogram. Therefore, WXYZ is not a rectangle. ANSWER: So the diagonals are congruent. Thus, ABCD is a No; slope of , slope of , slope of rectangle.

ANSWER: , and slope of . Slope of Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a slope of , and slope of slope of parallelogram. , so the diagonals are , so WXYZ is not a parallelogram. Therefore, congruent. Thus, ABCD is a rectangle. WXYZ is not a rectangle.

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance AD = 2 feet, and , find each eSolutionsFormulaManual - Powered by Cognero Page 3 measure. SOLUTION:

10. BC SOLUTION: The opposite sides of a rectangle are parallel and First, Use the Distance formula to find the lengths of congruent. Therefore, BC = AD = 2 ft. the sides of the quadrilateral. ANSWER: 2 ft

11. DB so ABCD is a SOLUTION:

parallelogram. All four angles of a rectangle are right angles. So, A parallelogram is a rectangle if the diagonals are is a right triangle. By the Pythagorean congruent. Use the Distance formula to find the 2 2 2 lengths of the diagonals. Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a ANSWER: 6.3 ft parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. 12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

By the Vertical Angle Theorem,

FENCING X-braces are also used to provide ANSWER: support in rectangular fencing. If AB = 6 feet, 50 AD = 2 feet, and , find each measure. 13. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, All four angles of a rectangle are right angles.

10. BC SOLUTION: ANSWER: The opposite sides of a rectangle are parallel and 25 congruent. Therefore, BC = AD = 2 ft. CCSS REGULARITY Quadrilateral WXYZ is a ANSWER: rectangle. 2 ft

11. DB SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 14. If ZY = 2x + 3 and WX = x + 4, find WX. 2 2 2 Theorem, DB = BC + DC . SOLUTION: BC = AD = 2 The opposite sides of a rectangle are parallel and DC = AB = 6 congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5. ANSWER: ANSWER: 6.3 ft 5 12. SOLUTION: 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, is an isosceles The diagonals of a rectangle bisect each other. So, triangle. PY = WP. Then,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. By the Vertical Angle Theorem,

ANSWER: 50 ANSWER: 13. 43 SOLUTION: 16. If , find The diagonals of a rectangle are congruent and . bisect each other. So, is an isosceles SOLUTION: triangle. Then, All four angles of a rectangle are right angles. So, All four angles of a rectangle are right angles.

ANSWER: 25 ANSWER: CCSS REGULARITY Quadrilateral WXYZ is a rectangle. 39 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY.

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: The opposite sides of a rectangle are parallel and Then ZP = 4(7) – 9 = 19. congruent. So, ZY = WX. Therefore, ZX = 2(ZP) = 38. 2x + 3 = x + 4 x = 1 ANSWER: Use the value of x to find WX. 38 WX = 1 + 4 = 5. 18. If , find ANSWER: . 5 SOLUTION: All four angles of a rectangle are right angles. So, 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP.

The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 48

19. If , find .

ANSWER: SOLUTION: 43 All four angles of a rectangle are right angles. So, is a right triangle. Then, 16. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: 46 ANSWER: PROOF Write a two-column proof. 39 20. Given: ABCD is a rectangle. Prove: 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY.

SOLUTION: You need to walk through the proof step by step. Then ZP = 4(7) – 9 = 19. Look over what you are given and what you need to Therefore, ZX = 2(ZP) = 38. prove. Here, you are given ABCD is a rectangle. You need to prove . Use the ANSWER: properties that you have learned about rectangles to 38 walk through the proof.

18. If , find Given: ABCD is a rectangle. . Prove: Proof: SOLUTION: Statements (Reasons) All four angles of a rectangle are right angles. So, 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS)

ANSWER: ANSWER: 48 Proof: Statements (Reasons) 19. If , find 1. ABCD is a rectangle. (Given) . 2. ABCD is a parallelogram. (Def. of rectangle)

SOLUTION: 3. (Opp. sides of a .) All four angles of a rectangle are right angles. So, 4. (Refl. Prop.) is a right triangle. Then, 5. (Diagonals of a rectangle are .) 6. (SSS)

21. Given: QTVW is a rectangle.

Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46. Prove: ANSWER: 46 SOLUTION: You need to walk through the proof step by step. PROOF Write a two-column proof. Look over what you are given and what you need to 20. Given: ABCD is a rectangle. prove. Here, you are given QTVW is a rectangle and Prove: . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: QTVW is a rectangle.

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to Prove: walk through the proof. Proof: Statements (Reasons) Given: ABCD is a rectangle. 1. QTVW is a rectangle; .(Given) Prove: 2. QTVW is a parallelogram. (Def. of rectangle) Proof: Statements (Reasons) 3. (Opp sides of a .) 1. ABCD is a rectangle. (Given) 4. are right angles.(Def. of rectangle) 2. ABCD is a parallelogram. (Def. of rectangle) 5. (All rt .)

3. (Opp. sides of a .) 6. QR = ST (Def. of segs.)

4. (Refl. Prop.) 7. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 8. RS = RS (Def. of segs.) 6. (SSS) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) ANSWER: 11. QS = RT (Subst.) Proof: 12. (Def. of segs.) Statements (Reasons) 1. ABCD is a rectangle. (Given) 13. (SAS) 2. ABCD is a parallelogram. (Def. of rectangle) ANSWER:

3. (Opp. sides of a .) Proof: 4. (Refl. Prop.) Statements (Reasons) 5. (Diagonals of a rectangle are .) 1. QTVW is a rectangle; .(Given) 6. (SSS) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 21. Given: QTVW is a rectangle. 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) Prove: 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) SOLUTION: 12. (Def. of segs.) You need to walk through the proof step by step. 13. (SAS) Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and COORDINATE GEOMETRY Graph each . You need to prove . Use quadrilateral with the given vertices. Determine the properties that you have learned about rectangles whether the figure is a rectangle. Justify your to walk through the proof. answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula Given: QTVW is a rectangle. SOLUTION:

Prove: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle)

3. (Opp sides of a .) Use the slope formula to find the slope of the sides of 4. are right angles.(Def. of rectangle) the quadrilateral. 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS) The slopes of each pair of opposite sides are equal. ANSWER: So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. Proof: The products of the slopes of the adjacent sides are Statements (Reasons) –1. So, any two adjacent sides are perpendicular to 1. QTVW is a rectangle; .(Given) each other. That is, all the four angles are right 2. QTVW is a parallelogram. (Def. of rectangle) angles. Therefore, WXYZ is a rectangle. 3. (Opp sides of a .) ANSWER: 4. are right angles.(Def. of rectangle) 5. (All rt .) Yes; slope of slope of , slope 6. QR = ST (Def. of segs.) of slope of . So WXYZ is a 7. (Refl. Prop.) parallelogram. The product of the slopes of 8. RS = RS (Def. of segs.) consecutive sides is –1, so the consecutive sides are 9. QR + RS = RS + ST (Add. prop.) perpendicular and form right angles. Thus, WXYZ is a 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) rectangle. 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. so JKLM is a The products of the slopes of the adjacent sides are parallelogram. –1. So, any two adjacent sides are perpendicular to A parallelogram is a rectangle if the diagonals are each other. That is, all the four angles are right congruent. Use the Distance formula to find the angles. Therefore, WXYZ is a rectangle. lengths of the diagonals.

ANSWER:

Yes; slope of slope of , slope So the diagonals are not congruent. Thus, JKLM is not a rectangle. of slope of . So WXYZ is a parallelogram. The product of the slopes of ANSWER: consecutive sides is –1, so the consecutive sides are No; so JKLM is a perpendicular and form right angles. Thus, WXYZ is a parallelogram; so rectangle. the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION: SOLUTION:

First, Use the Distance formula to find the lengths of First, use the Distance Formula to find the lengths of the sides of the quadrilateral. the sides of the quadrilateral.

so JKLM is a so QRST is a parallelogram. parallelogram. A parallelogram is a rectangle if the diagonals are A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the congruent. Use the Distance Formula to find the lengths of the diagonals. lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is So the diagonals are congruent. Thus, QRST is a not a rectangle. rectangle.

ANSWER: ANSWER: No; so JKLM is a Yes; so QRST is a parallelogram; so parallelogram. so the diagonals are the diagonals are not congruent. Thus, JKLM is not a congruent. QRST is a rectangle. rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION: 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, use the Distance Formula to find the lengths of First, use the slope formula to find the lengths of the

the sides of the quadrilateral. sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals. The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a So the diagonals are congruent. Thus, QRST is a parallelogram. rectangle. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. ANSWER: Therefore, GHJK is not a rectangle. Yes; so QRST is a ANSWER: parallelogram. so the diagonals are No; slope of slope of and slope of congruent. QRST is a rectangle. slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

26. SOLUTION: First, use the slope formula to find the lengths of the sides of the quadrilateral. All four angles of a rectangle are right angles. So,

ANSWER: 50

27. SOLUTION: The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. The measures of angles 2 and 3 are equal as they Therefore, the quadrilateral GHJK is a are alternate interior angles. parallelogram. Since the diagonals of a rectangle are congruent and None of the adjacent sides have slopes whose bisect each other, the triangle with the angles 3, 7 product is –1. So, the angles are not right angles. and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, GHJK is not a rectangle. Therefore, m∠7 = m∠2 = 40.

ANSWER: ANSWER: 40 No; slope of slope of and slope of 28. slope of . So, GHJK is a parallelogram. The product of the slopes of SOLUTION: consecutive sides ≠ –1, so the consecutive sides are The measures of angles 2 and 3 are equal as they not perpendicular. Thus, GHJK is not a rectangle. are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they are alternate interior angles. Quadrilateral ABCD is a rectangle. Find each Therefore, m∠4 = m∠1 = 50. measure if . Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: 26. 80 SOLUTION: 30. All four angles of a rectangle are right angles. So, SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 50 The measures of angles 1 and 4 are equal as they are alternate interior angles. 27. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 The measures of angles 2 and 3 are equal as they and 6 is an isosceles triangle. Therefore, m∠6 = are alternate interior angles. m∠4 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 ANSWER: and 8 is an isosceles triangle. So, m∠7 = m∠3. 50 Therefore, m∠7 = m∠2 = 40. 31. ANSWER: SOLUTION: 40 The measures of angles 2 and 3 are equal as they 28. are alternate interior angles. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 The measures of angles 2 and 3 are equal as they and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. are alternate interior angles. The sum of the three angles of a triangle is 180. Therefore, m∠3 = m∠2 = 40. Therefore, m∠8 = 180 – (40 + 40) = 100. ANSWER: ANSWER: 40 100

29. 32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one SOLUTION: shown. To what length should she cut the metal All four angles of a rectangle are right angles. So, supports in order for the bookshelf to be square, which means that the angles formed by the shelves and the vertical supports are all right angles? Explain your reasoning. The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: SOLUTION: 80 In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we 30. know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find SOLUTION: the lengths of the metal supports. All four angles of a rectangle are right angles. So, Let x be the length of each support. Express 3 feet as 36 inches.

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and Therefore, the metal supports should each be 39 bisect each other, the triangle with the angles 4, 5 inches long or 3 feet 3 inches long. and 6 is an isosceles triangle. Therefore, m∠6 = ANSWER: m∠4 = 50. 3 ft 3 in.; Sample answer: In order for the bookshelf ANSWER: to be square, the lengths of all of the metal supports 50 should be equal. Since I knew the length of the bookshelves and the distance between them, I used 31. the Pythagorean Theorem to find the lengths of the metal supports. SOLUTION: The measures of angles 2 and 3 are equal as they PROOF Write a two-column proof. are alternate interior angles. 33. Theorem 6.13 Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 You need to walk through the proof step by step.

and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. Look over what you are given and what you need to The sum of the three angles of a triangle is 180. prove. Here, you are given WXYZ is a rectangle with Therefore, m∠8 = 180 – (40 + 40) = 100. diagonals . You need to prove ANSWER: . Use the properties that you have learned 100 about rectangles to walk through the proof.

32. CCSS MODELING Jody is building a new Given: WXYZ is a rectangle with diagonals bookshelf using wood and metal supports like the one shown. To what length should she cut the metal . supports in order for the bookshelf to be square, Prove: which means that the angles formed by the shelves and the vertical supports are all right angles? Explain your reasoning.

Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .)

3. (Refl. Prop.) 4. are right angles. (Def. of SOLUTION: rectangle) In order for the bookshelf to be square, the lengths of 5. (All right .) all of the metal supports should be equal. Since we know the length of the bookshelves and the distance 6. (SAS) between them, use the Pythagorean Theorem to find 7. (CPCTC) the lengths of the metal supports. Let x be the length of each support. Express 3 feet ANSWER: as 36 inches. Given: WXYZ is a rectangle with diagonals . Prove:

Therefore, the metal supports should each be 39 inches long or 3 feet 3 inches long.

ANSWER: Proof: 3 ft 3 in.; Sample answer: In order for the bookshelf 1. WXYZ is a rectangle with diagonals . to be square, the lengths of all of the metal supports (Given) should be equal. Since I knew the length of the 2. (Opp. sides of a .) bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the 3. (Refl. Prop.) metal supports. 4. are right angles. (Def. of rectangle) PROOF Write a two-column proof. 5. (All right .) 33. Theorem 6.13 6. (SAS) SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with 34. Theorem 6.14 diagonals . You need to prove SOLUTION: . Use the properties that you have learned You need to walk through the proof step by step. about rectangles to walk through the proof. Look over what you are given and what you need to prove. Here, you are given Given: WXYZ is a rectangle with diagonals . You need to . prove that WXYZ is a rectangle.. Use the properties Prove: that you have learned about rectangles to walk through the proof.

Given: Prove: WXYZ is a rectangle.

Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) Proof: 3. (Refl. Prop.) 1. (Given) 4. are right angles. (Def. of 2. (SSS) rectangle) 3. (CPCTC) 5. (All right .) 4. (Def. of ) 6. (SAS) 5. WXYZ is a parallelogram. (If both pairs of opp.

7. (CPCTC) sides are , then quad. is .) ANSWER: 6. are supplementary. (Cons. Given: WXYZ is a rectangle with diagonals are suppl.)

. 7. (Def of suppl.) 8. are right angles. (If Prove: 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

Proof: ANSWER: 1. WXYZ is a rectangle with diagonals . Given: (Given) Prove: WXYZ is a rectangle. 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle)

5. (All right .) Proof: 6. (SAS) 1. (Given) 7. (CPCTC) 2. (SSS) 3. (CPCTC) 34. Theorem 6.14 4. (Def. of ) SOLUTION: 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) You need to walk through the proof step by step. Look over what you are given and what you need to 6. are supplementary. (Cons. prove. Here, you are given are suppl.) . You need to 7. (Def of suppl.) prove that WXYZ is a rectangle.. Use the properties 8. are right angles. (If that you have learned about rectangles to walk 2 and suppl., each is a rt. .)

through the proof. 9. are right angles. (If a has 1

rt. , it has 4 rt. .) Given: 10. WXYZ is a rectangle. (Def. of rectangle) Prove: WXYZ is a rectangle. PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION: Proof: You need to walk through the proof step by step. 1. (Given) Look over what you are given and what you need to 2. (SSS) prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. 3. (CPCTC) Let ABCD be a parallelogram with one right angle. 4. (Def. of ) Then use the properties of parallelograms to walk 5. WXYZ is a parallelogram. (If both pairs of opp. through the proof. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 ABCD is a parallelogram, and is a right angle. rt. , it has 4 rt. .) Since ABCD is a parallelogram and has one right 10. WXYZ is a rectangle. (Def. of rectangle) angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. ANSWER: ANSWER: Given: Prove: WXYZ is a rectangle.

ABCD is a parallelogram, and is a right angle. Proof: Since ABCD is a parallelogram and has one right 1. (Given) angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. 2. (SSS) 3. (CPCTC) 36. If a quadrilateral has four right angles, then it is a 4. (Def. of ) rectangle. 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) SOLUTION: 6. are supplementary. (Cons. You need to walk through the proof step by step. Look over what you are given and what you need to are suppl.) prove. Here, you are given a quadrilateral with four 7. (Def of suppl.) right angles. You need to prove that it is a rectangle. 8. are right angles. (If Let ABCD be a quadrilateral with four right angles. 2 and suppl., each is a rt. .) Then use the properties of parallelograms and

9. are right angles. (If a has 1 rectangles to walk through the proof.

rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. ABCD is a quadrilateral with four right angles. SOLUTION: ABCD is a parallelogram because both pairs of You need to walk through the proof step by step. opposite angles are congruent. By definition of a Look over what you are given and what you need to rectangle, ABCD is a rectangle. prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. ANSWER: Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

Step 2: ABCD is a parallelogram, and is a right angle. Place the compass on point P. Using the Since ABCD is a parallelogram and has one right same compass setting, draw an arc to the left and angle, then it has four right angles. So by the right of P that intersects . definition of a rectangle, ABCD is a rectangle. Step 3: Open the compass to a setting greater than 36. If a quadrilateral has four right angles, then it is a the distance from P to either point of intersection. rectangle. Place the compass on each point of intersection and SOLUTION: draw arcs that intersect above . You need to walk through the proof step by step. Look over what you are given and what you need to Step 4: Draw line a through point P and the prove. Here, you are given a quadrilateral with four intersection of the arcs drawn in step 3. right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. Step 5: Repeat steps 2 and 3 using point Q. Then use the properties of parallelograms and rectangles to walk through the proof. Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. 37. CONSTRUCTION Construct a rectangle using the Since and , the measure of construction for congruent segments and the angles P and Q is 90 degrees. Lines that are construction for a line perpendicular to another line perpendicular to the same line are parallel, so through a point on the line. Justify each step of the . The same compass setting was used to construction. locate points R and S, so . If one pair of SOLUTION: opposite sides of a quadrilateral is both parallel and Sample answer: congruent, then the quadrilateral is a parallelogram. Step 1: Graph a line and place points P and Q on the A parallelogram with right angles is a rectangle. line. Thus, PRSQ is a rectangle.

Step 2: Place the compass on point P. Using the ANSWER: same compass setting, draw an arc to the left and

right of P that intersects . Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that Step 3: Open the compass to a setting greater than are perpendicular to the same line are parallel, so the distance from P to either point of intersection. Place the compass on each point of intersection and . The same compass setting was used to draw arcs that intersect above . locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and Step 4: Draw line a through point P and the congruent, then the quadrilateral is a parallelogram. intersection of the arcs drawn in step 3. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S. 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for Step 9: Draw . painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then the end zone is a rectangle. Since and , the measure of angles P and Q is 90 degrees. Lines that are ANSWER: perpendicular to the same line are parallel, so Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the . The same compass setting was used to diagonals and both pairs of opposite sides are locate points R and S, so . If one pair of congruent, then the end zone is a rectangle. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. ALGEBRA Quadrilateral WXYZ is a rectangle. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

ANSWER: 39. If XW = 3, WZ = 4, and XZ = b, find YW.

SOLUTION: Sample answer: Since and , the The diagonals of a rectangle are congruent to each measure of angles P and Q is 90 degrees. Lines that other. So, YW = XZ = b. All four angles of a are perpendicular to the same line are parallel, so rectangle are right angles. So, is a right . The same compass setting was used to 2 2 triangle. By the Pythagorean Theorem, XZ = XW + 2 locate points R and S, so . If one pair of WZ . opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Therefore, YW = 5.

ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: The diagonals of a rectangle are congruent to each other. So, WY = XZ = 2c. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XY2 + 2 38. SPORTS The end zone of a football field is 160 feet ZY . wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Therefore, WY = 2(5) = 10. Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should ANSWER: measure the diagonals of the end zone and both 10 pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a 41. SIGNS The sign below is in the foyer of Nyoko’s parallelogram. If the diagonals are congruent, then school. Based on the dimensions given, can Nyoko the end zone is a rectangle. be sure that the sign is a rectangle? Explain your reasoning. ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

SOLUTION:

Both pairs of opposite sides are congruent, so the 39. If XW = 3, WZ = 4, and XZ = b, find YW. sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the SOLUTION: diagonals could be proven to be congruent, or if the The diagonals of a rectangle are congruent to each angles could be proven to be right angles, then we other. So, YW = XZ = b. All four angles of a could prove that the sign was a rectangle. rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XW 2 + ANSWER: 2 No; sample answer: Both pairs of opposite sides are WZ . congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

Therefore, YW = 5. PROOF Write a coordinate proof of each statement. ANSWER: 42. The diagonals of a rectangle are congruent. 5 SOLUTION: 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let SOLUTION: the length of the bases be a units and the height be b The diagonals of a rectangle are congruent to each units. Then the rest of the vertices are A(0, b), B(a, other. So, WY = XZ = 2c. All four angles of a b), and C(a, 0). You need to walk through the proof rectangle are right angles. So, is a right step by step. Look over what you are given and what 2 2 you need to prove. Here, you are given ABCD is a triangle. By the Pythagorean Theorem, XZ = XY + 2 rectangle and you need to prove . Use the ZY . properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove:

Therefore, WY = 2(5) = 10.

ANSWER: 10

41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your Proof: reasoning. Use the Distance Formula to find the lengths of the diagonals.

The diagonals, have the same length, so they are congruent.

ANSWER: SOLUTION: Given: ABCD is a rectangle.

Both pairs of opposite sides are congruent, so the Prove: sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no Proof: measure is given that can be used to prove that it is a Use the Distance Formula to find rectangle. and . PROOF Write a coordinate proof of each have the same length, so they are statement. congruent. 42. The diagonals of a rectangle are congruent. SOLUTION: 43. If the diagonals of a parallelogram are congruent, then it is a rectangle. Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let SOLUTION: the length of the bases be a units and the height be b Begin by positioning parallelogram ABCD on a units. Then the rest of the vertices are A(0, b), B(a, coordinate plane. Place vertex A at the origin. Let b), and C(a, 0). You need to walk through the proof the length of the bases be a units and the height be c step by step. Look over what you are given and what units. Then the rest of the vertices are B(a, 0), C(b + you need to prove. Here, you are given ABCD is a a, c), and D(b, c). You need to walk through the rectangle and you need to prove . Use the proof step by step. Look over what you are given properties that you have learned about rectangles to and what you need to prove. Here, you are given walk through the proof. ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you Given: ABCD is a rectangle. have learned about rectangles to walk through the Prove: proof.

Given: ABCD and Prove: ABCD is a rectangle.

Proof: Use the Distance Formula to find the lengths of the

diagonals. Proof:

The diagonals, have the same length, so AC = BD. So, they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle. Proof: Use the Distance Formula to find ANSWER: and . Given: ABCD and have the same length, so they are Prove: ABCD is a rectangle. congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION:

Begin by positioning parallelogram ABCD on a Proof: coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given But AC = BD. So, and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special Proof: parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and AC = BD. So, label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

ANSWER:

Given: ABCD and Prove: ABCD is a rectangle.

b. Use a protractor to measure each angle listed in Proof: the table.

c. Sample answer: Each of the angles listed in the

But AC = BD. So, table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: a. Sample answer:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . DAB is ∠ a right angle and ABCD is a rectangle..

label the intersections R. b. TABULAR Use a protractor to measure the c. Sample answer: The diagonals of a parallelogram appropriate angles and complete the table below. with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, c. VERBAL Make a conjecture about the diagonals Find the of a parallelogram with four congruent sides. values of x and y. SOLUTION: a. Sample answer:

SOLUTION: All four angles of a rectangle are right angles. So,

b. Use a protractor to measure each angle listed in The diagonals of a rectangle are congruent and the table. bisect each other. So, is an isosceles triangle with So,

ANSWER: So, By the Vertical Angle Theorem, a. Sample answer:

ANSWER: x = 6, y = −10

quadrilateral, two of the angles are formed by a c. Sample answer: The diagonals of a parallelogram single vertex of a triangle. with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

SOLUTION: All four angles of a rectangle are right angles. So, So, Tamika is correct.

ANSWER:

Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of The diagonals of a rectangle are congruent and the angles are formed by a single vertex of a triangle. bisect each other. So, is an isosceles triangle In order for the quadrilateral to be a rectangle, one of with the angles in the congruent triangles has to be a right So, angle.

ANSWER: SOLUTION: x = 6, y = −10 Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the 46. CCSS CRITIQUE Parker says that any two lines intersect the line m at the points W, X, Y, and Z. congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right ANSWER: angle. 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer:

So, Tamika is correct. A rectangle has 4 right angles and each pair of ANSWER: opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other Tamika; Sample answer: When two congruent and perpendicular to lines parallel to the y-axis. triangles are arranged to form a quadrilateral, two of Selecting y = 0 and x = 0 will form 2 sides of the the angles are formed by a single vertex of a triangle. rectangle. The lines y = 4 and x = 6 form the other 2 In order for the quadrilateral to be a rectangle, one of sides. the angles in the congruent triangles has to be a right

angle. x = 0, x = 6, y = 0, y = 4; 47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines?

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the

slope of DC is 0. Since a pair of sides of the SOLUTION: quadrilateral is both parallel and congruent, by Let the lines n, p, q, and r intersect the line l at the Theorem 6.12, the quadrilateral is a parallelogram. points A, B, C, and D respectively. Similarly, let the Since is horizontal and is vertical, the lines lines intersect the line m at the points W, X, Y, and Z. are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6 The length of is 6 − 0 or 6 units and the length of 48. OPEN ENDED Write the equations of four lines is 6 – 0 or 6 units. The slope of AB is 0 and the having intersections that form the vertices of a slope of DC is 0. Since a pair of sides of the rectangle. Verify your answer using coordinate quadrilateral is both parallel and congruent, by geometry. Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines SOLUTION: are perpendicular and the measure of the angle they Sample answer: form is 90. By Theorem 6.6, if a parallelogram has A rectangle has 4 right angles and each pair of one right angle, it has four right angles. Therefore, by opposite sides is parallel and congruent. Lines that definition, the parallelogram is a rectangle. run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. 49. WRITING IN MATH Explain why all rectangles Selecting y = 0 and x = 0 will form 2 sides of the are parallelograms, but all parallelograms are not rectangle. The lines y = 4 and x = 6 form the other 2 rectangles. sides. SOLUTION: x = 0, x = 6, y = 0, y = 4; Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms.

Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

ANSWER: The length of is 6 − 0 or 6 units and the length of Sample answer: All rectangles are parallelograms is 6 – 0 or 6 units. The slope of AB is 0 and the because, by definition, both pairs of opposite sides slope of DC is 0. Since a pair of sides of the are parallel. Parallelograms with right angles are quadrilateral is both parallel and congruent, by rectangles, so some parallelograms are rectangles, Theorem 6.12, the quadrilateral is a parallelogram. but others with non-right angles are not. Since is horizontal and is vertical, the lines 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = are perpendicular and the measure of the angle they 13, what values of x and y make parallelogram form is 90. By Theorem 6.6, if a parallelogram has FGHJ a rectangle? one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4; A x = 3, y = 4 B x = 4, y = 3 C x = 7, y = 8 D x = 8, y = 7 SOLUTION: The opposite sides of a parallelogram are congruent. So, FJ = GH. –3x + 5y = 11 FJ = GH

If the diagonals of a parallelogram are congruent, The length of is 6 − 0 or 6 units and the length of then it is a rectangle. Since FGHJ is a parallelogram, is 6 – 0 or 6 units. The slope of AB is 0 and the the diagonals bisect each other. So, if it is a slope of DC is 0. Since a pair of sides of the rectangle, then will be an isosceles triangle quadrilateral is both parallel and congruent, by with That is, 3x + y = 13. Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines Add the two equations to eliminate the x-term and are perpendicular and the measure of the angle they solve for y. form is 90. By Theorem 6.6, if a parallelogram has –3x + 5y = 11 one right angle, it has four right angles. Therefore, by 3x + y = 13 definition, the parallelogram is a rectangle. 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6. 49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not Use the value of y to find the value of x. rectangles. 3x + 4 = 13 Substitute. SOLUTION: 3x = 9 Subtract 4 from each side. Sample answer: A parallelogram is a quadrilateral in x = 3 Divide each side by 3. which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs So, x = 3, y = 4, the correct choice is A. of opposite sides parallel. So rectangles are always ANSWER: parallelograms. A Parallelograms with right angles are rectangles, so ALGEBRA some parallelograms are rectangles, but others with 51. A rectangular playground is surrounded non-right angles are not. by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following ANSWER: equations could be used to find r, the shorter side of Sample answer: All rectangles are parallelograms the playground? because, by definition, both pairs of opposite sides F 10r + r = 80 are parallel. Parallelograms with right angles are G 4r + 10 = 80 rectangles, so some parallelograms are rectangles, H r(r + 10) = 80 but others with non-right angles are not. J 2(r + 10) + 2r = 80 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = SOLUTION: 13, what values of x and y make parallelogram The formula to find the perimeter of a rectangle is FGHJ a rectangle? given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. Therefore, the correct choice is J.

ANSWER: A x = 3, y = 4 J B x = 4, y = 3 C x = 7, y = 8 52. SHORT RESPONSE What is the measure of D x = 8, y = 7 ? SOLUTION: The opposite sides of a parallelogram are congruent. So, FJ = GH. –3x + 5y = 11 FJ = GH

If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, SOLUTION: the diagonals bisect each other. So, if it is a The diagonals of a rectangle are congruent and rectangle, then will be an isosceles triangle bisect each other. So, is an isosceles triangle with That is, 3x + y = 13. with So, Add the two equations to eliminate the x-term and By the Exterior Angle Theorem, solve for y. –3x + 5y = 11 3x + y = 13 ANSWER: 6y = 24 Add to eliminate x-term. 112 y = 4 Divide each side by 6. 53. SAT/ACT If p is odd, which of the following must Use the value of y to find the value of x. also be odd? 3x + 4 = 13 Substitute. A 2p 3x = 9 Subtract 4 from each side. B 2p + 2 x = 3 Divide each side by 3. C

the playground? This can be rewritten as 2(p + 1) which is also a F 10r + r = 80 multiple of 2 so it is even. G 4r + 10 = 80 H r(r + 10) = 80 Choice C J 2(r + 10) + 2r = 80 If p is odd , is not a whole number. SOLUTION: Choice D 2p – 2 The formula to find the perimeter of a rectangle is This can be rewritten as 2(p - 1) which is also a given by P = 2(l + w). The perimeter is given to be 8 multiple of 2 so it is even. ft. The width is r and the length is r + 10. So, 2r + 2 Choice E p + 2 (r + 10) = 80. Adding 2 to an odd number results in an odd Therefore, the correct choice is J. number. This is odd. The correct choice is E. ANSWER: J ANSWER: E 52. SHORT RESPONSE What is the measure of ? ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

54. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and The opposite sides of a parallelogram are congruent. bisect each other. So, is an isosceles triangle with So,

By the Exterior Angle Theorem, In a parallelogram, consecutive interior angles are supplementary. ANSWER: 112

53. SAT/ACT If p is odd, which of the following must also be odd? A 2p B 2p + 2 ANSWER: x = 2, y = 41 C D E p + 2 SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. 55. Choice A 2p This is a multiple of 2 so it is even. SOLUTION: Choice B 2p + 2 The opposite sides of a parallelogram are congruent. This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

Choice C The opposite sides of a parallelogram are parallel to If p is odd , is not a whole number. each other. So, the angles with the measures (3y + Choice D 2p – 2 3)° and (4y – 19)° are alternate interior angles and

This can be rewritten as 2(p - 1) which is also a hence they are equal.

multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. The correct choice is E. ANSWER: x = 8, y = 22 ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram. 56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. 54. Solve the first equation to find the value of y. SOLUTION: 2y = 14 The opposite sides of a parallelogram are congruent. y = 7 Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x In a parallelogram, consecutive interior angles are 2 = x supplementary. ANSWER: x = 2, y = 7

57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C ANSWER: (4, –2), and D(–1, –1). x = 2, y = 41 SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

55. Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5). SOLUTION: The opposite sides of a parallelogram are congruent. ANSWER: (2.5, 0.5)

Refer to the figure. The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

ANSWER: 58. If , name two congruent angles. x = 8, y = 22 SOLUTION: Since is an isosceles triangle. So,

ANSWER: 56. SOLUTION: The diagonals of a parallelogram bisect each other. 59. If , name two congruent segments. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. SOLUTION: Solve the first equation to find the value of y. In a triangle with two congruent angles, the sides 2y = 14 opposite to the congruent angles are also congruent. y = 7 Use the value of y in the second equation and solve So, .

for x. ANSWER: 7 + 3 = 2x + 6 4 = 2x 2 = x 60. If , name two congruent segments. ANSWER: SOLUTION: x = 2, y = 7 In a triangle with two congruent angles, the sides 57. COORDINATE GEOMETRY Find the opposite to the congruent angles are also congruent. coordinates of the intersection of the diagonals of So, . parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). ANSWER:

SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. ANSWER:

If feet, PS = 7 feet, and , find each measure. 3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Therefore,

1. QR ANSWER: SOLUTION: 33.5 The opposite sides of a rectangle are parallel and 4. congruent. Therefore, QR = PS = 7 ft. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 7 ft bisect each other. So, is an isosceles triangle. Then, 2. SQ By the Vertical Angle Theorem,

SOLUTION: The diagonals of a rectangle bisect each other. So, Therefore,

ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

ANSWER:

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: 3. The diagonals of a rectangle are congruent to each other. So, FD = EG. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, Use the value of x to find EG. By the Exterior Angle Theorem, EG = 6 + 5 = 11

ANSWER: Therefore, 11

ANSWER: 6. If , 33.5 find . 4. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore, ANSWER: ANSWER: 51 56.5 7. PROOF If ABDE is a rectangle and , ALGEBRA Quadrilateral DEFG is a rectangle. prove that .

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION:

The diagonals of a rectangle are congruent to each other. So, FD = EG. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and Use the value of x to find EG. . You need to prove . Use the EG = 6 + 5 = 11 properties that you have learned about rectangles to walk through the proof. ANSWER: 11 Given: ABDE is a rectangle; Prove: 6. If , find . SOLUTION: All four angles of a rectangle are right angles. So,

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle)

ANSWER: 3. (Opp. sides of a .) 51 4. are right angles.(Def. of rectangle) 5. (All rt .) 7. PROOF If ABDE is a rectangle and , 6. (SAS)

prove that . 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and Statements(Reasons) . You need to prove . Use the 1. ABDE is a rectangle; . properties that you have learned about rectangles to (Given) walk through the proof. 2. ABDE is a parallelogram. (Def. of

rectangle)

Given: ABDE is a rectangle; 3. (Opp. sides of a .)

Prove: 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine Statements(Reasons) whether the figure is a rectangle. Justify your answer using the indicated formula. 1. ABDE is a rectangle; . 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula (Given) 2. ABDE is a parallelogram. (Def. of SOLUTION: rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle;

Prove: Use the slope formula to find the slope of the sides of the quadrilateral.

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of

rectangle) There are no parallel sides, so WXYZ is not a 3. (Opp. sides of a .) parallelogram. Therefore, WXYZ is not a rectangle. 4. are right angles.(Def. of rectangle) ANSWER: 5. (All rt .) 6. (SAS) No; slope of , slope of , slope of 7. (CPCTC) , and slope of . Slope of COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine slope of , and slope of slope of whether the figure is a rectangle. Justify your , so WXYZ is not a parallelogram. Therefore, answer using the indicated formula. WXYZ is not a rectangle. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Use the slope formula to find the slope of the sides of Formula the quadrilateral. SOLUTION:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER: First, Use the Distance formula to find the lengths of the sides of the quadrilateral. No; slope of , slope of , slope of

, and slope of . Slope of

slope of , and slope of slope of so ABCD is a , so WXYZ is not a parallelogram. Therefore, parallelogram. WXYZ is not a rectangle. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure. First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the 10. BC lengths of the diagonals. SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. So the diagonals are congruent. Thus, ABCD is a rectangle. ANSWER: 2 ft ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a 11. DB parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

6-4 Rectangles ANSWER: 6.3 ft FENCING X-braces are also used to provide 12. support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each SOLUTION: measure. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

By the Vertical Angle Theorem,

10. BC ANSWER: 50 SOLUTION: The opposite sides of a rectangle are parallel and 13. congruent. Therefore, BC = AD = 2 ft. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 2 ft bisect each other. So, is an isosceles triangle. 11. DB Then, All four angles of a rectangle are right angles. SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . ANSWER: BC = AD = 2 DC = AB = 6 25 CCSS REGULARITY Quadrilateral WXYZ is a rectangle.

ANSWER: 6.3 ft

12.

SOLUTION: 14. If ZY = 2x + 3 and WX = x + 4, find WX. The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, is an isosceles The opposite sides of a rectangle are parallel and triangle. congruent. So, ZY = WX. Then, 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5. By the Vertical Angle Theorem, ANSWER: 5 ANSWER: eSolutions50 Manual - Powered by Cognero 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. Page 4 13. SOLUTION: The diagonals of a rectangle bisect each other. So, SOLUTION: PY = WP. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. The diagonals of a rectangle are congruent and Then, bisect each other. So, is an isosceles triangle. All four angles of a rectangle are right angles.

ANSWER: ANSWER: 25 43 CCSS REGULARITY Quadrilateral WXYZ is a 16. If , find rectangle. . SOLUTION: All four angles of a rectangle are right angles. So,

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 2x + 3 = x + 4 ANSWER: x = 1 39 Use the value of x to find WX. WX = 1 + 4 = 5. 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. ANSWER: SOLUTION: 5 The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY.

15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: The diagonals of a rectangle bisect each other. So, Then ZP = 4(7) – 9 = 19. PY = WP. Therefore, ZX = 2(ZP) = 38.

ANSWER: The diagonals of a rectangle are congruent and 38 bisect each other. So, is an isosceles triangle. 18. If , find . SOLUTION: All four angles of a rectangle are right angles. So, ANSWER: 43 16. If , find .

SOLUTION: All four angles of a rectangle are right angles. So, ANSWER: 48

19. If , find . SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. Then, ANSWER: 39 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: The diagonals of a rectangle are congruent and Therefore, Since is a bisect each other. So, ZP = PY. transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: Then ZP = 4(7) – 9 = 19. 46 Therefore, ZX = 2(ZP) = 38. PROOF Write a two-column proof. ANSWER: 20. Given: ABCD is a rectangle. 38 Prove: 18. If , find . SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof. ANSWER:

48 Given: ABCD is a rectangle.

19. If , find Prove: Proof: . Statements (Reasons) SOLUTION: 1. ABCD is a rectangle. (Given)

All four angles of a rectangle are right angles. So, 2. ABCD is a parallelogram. (Def. of rectangle)

is a right triangle. Then, 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS)

ANSWER: Therefore, Since is a Proof: transversal of parallel sides and , alternate Statements (Reasons) interior angles WZX and ZXY are congruent. So, 1. ABCD is a rectangle. (Given) = = 46. 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) ANSWER:

46 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) PROOF Write a two-column proof. 6. (SSS) 20. Given: ABCD is a rectangle. Prove: 21. Given: QTVW is a rectangle.

SOLUTION: You need to walk through the proof step by step. Prove: Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. SOLUTION: You need to prove . Use the You need to walk through the proof step by step. properties that you have learned about rectangles to Look over what you are given and what you need to walk through the proof. prove. Here, you are given QTVW is a rectangle and . You need to prove . Use Given: ABCD is a rectangle. the properties that you have learned about rectangles Prove: to walk through the proof. Proof: Statements (Reasons) Given: QTVW is a rectangle. 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .)

6. (SSS) Prove: ANSWER: Proof:

Proof: Statements (Reasons) Statements (Reasons) 1. QTVW is a rectangle; .(Given) 1. ABCD is a rectangle. (Given) 2. QTVW is a parallelogram. (Def. of rectangle) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .)

3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 4. (Refl. Prop.) 5. (All rt .) 5. (Diagonals of a rectangle are .) 6. QR = ST (Def. of segs.) 6. (SSS) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 21. Given: QTVW is a rectangle. 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Prove: Proof: Statements (Reasons) SOLUTION:

You need to walk through the proof step by step. 1. QTVW is a rectangle; .(Given) Look over what you are given and what you need to 2. QTVW is a parallelogram. (Def. of rectangle) prove. Here, you are given QTVW is a rectangle and 3. (Opp sides of a .) . You need to prove . Use 4. are right angles.(Def. of rectangle) the properties that you have learned about rectangles 5. (All rt .) to walk through the proof. 6. QR = ST (Def. of segs.)

Given: QTVW is a rectangle. 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS) Prove: Proof: COORDINATE GEOMETRY Graph each Statements (Reasons) quadrilateral with the given vertices. Determine 1. QTVW is a rectangle; .(Given) whether the figure is a rectangle. Justify your answer using the indicated formula. 2. QTVW is a parallelogram. (Def. of rectangle) 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula 3. (Opp sides of a .) SOLUTION: 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Use the slope formula to find the slope of the sides of Proof: the quadrilateral. Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) The slopes of each pair of opposite sides are equal. 9. QR + RS = RS + ST (Add. prop.) So, the two pairs of opposite sides are parallel. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) Therefore, the quadrilateral WXYZ is a parallelogram. 11. QS = RT (Subst.) The products of the slopes of the adjacent sides are 12. (Def. of segs.) –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right

13. (SAS) angles. Therefore, WXYZ is a rectangle.

COORDINATE GEOMETRY Graph each ANSWER: quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your Yes; slope of slope of , slope answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula of slope of . So WXYZ is a parallelogram. The product of the slopes of SOLUTION: consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

ANSWER: First, Use the Distance formula to find the lengths of

Yes; slope of slope of , slope the sides of the quadrilateral.

of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle. so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER:

No; so JKLM is a parallelogram; so 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance the diagonals are not congruent. Thus, JKLM is not a Formula rectangle. SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula First, Use the Distance formula to find the lengths of SOLUTION: the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

First, use the Distance Formula to find the lengths of So the diagonals are not congruent. Thus, JKLM is the sides of the quadrilateral. not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so so QRST is a the diagonals are not congruent. Thus, JKLM is not a parallelogram. rectangle. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

First, use the slope formula to find the lengths of the sides of the quadrilateral. So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle.

ANSWER:

No; slope of slope of and slope of 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION: slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

First, use the slope formula to find the lengths of the sides of the quadrilateral.

Quadrilateral ABCD is a rectangle. Find each measure if .

The slopes of each pair of opposite sides are equal. 26. So, the two pairs of opposite sides are parallel. SOLUTION: Therefore, the quadrilateral GHJK is a All four angles of a rectangle are right angles. So,

parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle.

ANSWER: ANSWER: 50 No; slope of slope of and slope of

slope of . So, GHJK is a 27. parallelogram. The product of the slopes of SOLUTION: consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle. The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: 40

28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. Quadrilateral ABCD is a rectangle. Find each measure if . ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So, 26. SOLUTION: All four angles of a rectangle are right angles. So, The measures of angles 1 and 4 are equal as they are alternate interior angles.

Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 ANSWER: and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. 50 The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. 27. ANSWER: SOLUTION: 80 The measures of angles 2 and 3 are equal as they are alternate interior angles. 30. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 SOLUTION: and 8 is an isosceles triangle. So, m∠7 = m∠3. All four angles of a rectangle are right angles. So, Therefore, m∠7 = m∠2 = 40.

ANSWER: 40 The measures of angles 1 and 4 are equal as they are alternate interior angles. 28. Therefore, m∠4 = m∠1 = 50. SOLUTION: Since the diagonals of a rectangle are congruent and The measures of angles 2 and 3 are equal as they bisect each other, the triangle with the angles 4, 5 are alternate interior angles. and 6 is an isosceles triangle. Therefore, m∠6 = Therefore, m∠3 = m∠2 = 40. m∠4 = 50. ANSWER: ANSWER: 40 50

29. 31. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and

bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The measures of angles 1 and 4 are equal as they The sum of the three angles of a triangle is 180. are alternate interior angles. Therefore, m∠8 = 180 – (40 + 40) = 100. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 4, 5 100 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. 32. CCSS MODELING Jody is building a new Therefore, m∠5 = 180 – (50 + 50) = 80. bookshelf using wood and metal supports like the one shown. To what length should she cut the metal ANSWER: supports in order for the bookshelf to be square, 80 which means that the angles formed by the shelves and the vertical supports are all right angles? Explain 30. your reasoning. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they are alternate interior angles.

Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 In order for the bookshelf to be square, the lengths of and 6 is an isosceles triangle. Therefore, m∠6 = all of the metal supports should be equal. Since we m∠4 = 50. know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find ANSWER: the lengths of the metal supports. 50 Let x be the length of each support. Express 3 feet as 36 inches. 31. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and Therefore, the metal supports should each be 39 bisect each other, the triangle with the angles 3, 7 inches long or 3 feet 3 inches long. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. ANSWER: Therefore, m∠8 = 180 – (40 + 40) = 100. 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports ANSWER: should be equal. Since I knew the length of the 100 bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the 32. CCSS MODELING Jody is building a new metal supports. bookshelf using wood and metal supports like the one shown. To what length should she cut the metal PROOF Write a two-column proof. supports in order for the bookshelf to be square, 33. Theorem 6.13 which means that the angles formed by the shelves and the vertical supports are all right angles? Explain SOLUTION: your reasoning. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Let x be the length of each support. Express 3 feet Proof: as 36 inches. 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of Therefore, the metal supports should each be 39 rectangle) inches long or 3 feet 3 inches long. 5. (All right .) ANSWER: 6. (SAS) 3 ft 3 in.; Sample answer: In order for the bookshelf 7. (CPCTC) to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the ANSWER: bookshelves and the distance between them, I used Given: WXYZ is a rectangle with diagonals the Pythagorean Theorem to find the lengths of the metal supports. . Prove: PROOF Write a two-column proof. 33. Theorem 6.13 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with Proof: diagonals . You need to prove 1. WXYZ is a rectangle with diagonals . (Given) . Use the properties that you have learned

about rectangles to walk through the proof. 2. (Opp. sides of a .) 3. (Refl. Prop.) Given: WXYZ is a rectangle with diagonals 4. are right angles. (Def. of . rectangle) Prove: 5. (All right .) 6. (SAS) 7. (CPCTC)

34. Theorem 6.14 Proof: SOLUTION: 1. WXYZ is a rectangle with diagonals . You need to walk through the proof step by step. (Given) Look over what you are given and what you need to 2. (Opp. sides of a .) prove. Here, you are given 3. (Refl. Prop.) . You need to prove that WXYZ is a rectangle.. Use the properties 4. are right angles. (Def. of that you have learned about rectangles to walk

rectangle) through the proof. 5. (All right .) 6. (SAS) Given: 7. (CPCTC) Prove: WXYZ is a rectangle.

ANSWER: Given: WXYZ is a rectangle with diagonals . Prove: Proof: 1. (Given) 2. (SSS) 3. (CPCTC)

Proof: 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. 1. WXYZ is a rectangle with diagonals . sides are , then quad. is .) (Given) 6. are supplementary. (Cons. 2. (Opp. sides of a .) are suppl.)

3. (Refl. Prop.) 7. (Def of suppl.) 4. are right angles. (Def. of 8. are right angles. (If rectangle) 2 and suppl., each is a rt. .) 5. (All right .) 9. are right angles. (If a has 1 6. (SAS) rt. , it has 4 rt. .) 7. (CPCTC) 10. WXYZ is a rectangle. (Def. of rectangle)

ANSWER: 34. Theorem 6.14 Given: SOLUTION: Prove: WXYZ is a rectangle. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given . You need to prove that WXYZ is a rectangle.. Use the properties

that you have learned about rectangles to walk Proof:

through the proof. 1. (Given) Given: 2. (SSS) Prove: WXYZ is a rectangle. 3. (CPCTC) 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) Proof: 7. (Def of suppl.) 1. (Given) 8. are right angles. (If 2. (SSS) 2 and suppl., each is a rt. .) 3. (CPCTC) 9. are right angles. (If a has 1 4. (Def. of ) rt. , it has 4 rt. .) 5. WXYZ is a parallelogram. (If both pairs of opp. 10. WXYZ is a rectangle. (Def. of rectangle) sides are , then quad. is .) 6. are supplementary. (Cons. PROOF Write a paragraph proof of each statement.

are suppl.) 35. If a parallelogram has one right angle, then it is a 7. (Def of suppl.) rectangle. 8. are right angles. (If SOLUTION: 2 and suppl., each is a rt. .) You need to walk through the proof step by step. 9. are right angles. (If a has 1 Look over what you are given and what you need to rt. , it has 4 rt. .) prove. Here, you are given a parallelogram with one 10. WXYZ is a rectangle. (Def. of rectangle) right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. ANSWER: Then use the properties of parallelograms to walk Given: through the proof. Prove: WXYZ is a rectangle.

Proof: ABCD is a parallelogram, and is a right angle. 1. (Given) Since ABCD is a parallelogram and has one right 2. (SSS) angle, then it has four right angles. So by the

3. (CPCTC) definition of a rectangle, ABCD is a rectangle. 4. (Def. of ) ANSWER: 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.)

8. are right angles. (If ABCD is a parallelogram, and is a right angle. 2 and suppl., each is a rt. .) Since ABCD is a parallelogram and has one right 9. are right angles. (If a has 1 angle, then it has four right angles. So by the rt. , it has 4 rt. .) definition of a rectangle, ABCD is a rectangle. 10. WXYZ is a rectangle. (Def. of rectangle) 36. If a quadrilateral has four right angles, then it is a PROOF Write a paragraph proof of each rectangle. statement. 35. If a parallelogram has one right angle, then it is a SOLUTION: rectangle. You need to walk through the proof step by step. Look over what you are given and what you need to SOLUTION: prove. Here, you are given a quadrilateral with four You need to walk through the proof step by step. right angles. You need to prove that it is a rectangle. Look over what you are given and what you need to Let ABCD be a quadrilateral with four right angles. prove. Here, you are given a parallelogram with one Then use the properties of parallelograms and right angle. You need to prove that it is a rectangle. rectangles to walk through the proof. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

ANSWER:

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a ABCD is a parallelogram, and is a right angle. rectangle, ABCD is a rectangle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the 37. CONSTRUCTION Construct a rectangle using the definition of a rectangle, ABCD is a rectangle. construction for congruent segments and the construction for a line perpendicular to another line 36. If a quadrilateral has four right angles, then it is a through a point on the line. Justify each step of the rectangle. construction. SOLUTION: SOLUTION: You need to walk through the proof step by step. Sample answer: Look over what you are given and what you need to Step 1: Graph a line and place points P and Q on the prove. Here, you are given a quadrilateral with four line. right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. Step 2: Place the compass on point P. Using the Then use the properties of parallelograms and same compass setting, draw an arc to the left and

rectangles to walk through the proof. right of P that intersects .

Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Place the compass on each point of intersection and draw arcs that intersect above .

ABCD is a quadrilateral with four right angles. Step 4: Draw line a through point P and the ABCD is a parallelogram because both pairs of intersection of the arcs drawn in step 3. opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 5: Repeat steps 2 and 3 using point Q.

ANSWER: Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of Step 8: Using the same compass setting, put the opposite angles are congruent. By definition of a compass at Q and draw an arc above it that rectangle, ABCD is a rectangle. intersects line b. Label the point of intersection S.

37. CONSTRUCTION Construct a rectangle using the Step 9: Draw . construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. SOLUTION: Sample answer: Step 1: Graph a line and place points P and Q on the line.

Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and right of P that intersects . Since and , the measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. . The same compass setting was used to Place the compass on each point of intersection and locate points R and S, so . If one pair of draw arcs that intersect above . opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 4: Draw line a through point P and the A parallelogram with right angles is a rectangle. intersection of the arcs drawn in step 3. Thus, PRSQ is a rectangle.

Step 5: Repeat steps 2 and 3 using point Q. ANSWER:

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5. Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that Step 7: Using any compass setting, put the compass are perpendicular to the same line are parallel, so at P and draw an arc above it that intersects line a. . The same compass setting was used to Label the point of intersection R. locate points R and S, so . If one pair of Step 8: Using the same compass setting, put the opposite sides of a quadrilateral is both parallel and compass at Q and draw an arc above it that congruent, then the quadrilateral is a parallelogram. intersects line b. Label the point of intersection S. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Step 9: Draw .

38. SPORTS The end zone of a football field is 160 feet Since and , the measure of wide and 30 feet long. Kyle is responsible for angles P and Q is 90 degrees. Lines that are painting the field. He has finished the end zone. perpendicular to the same line are parallel, so Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a . The same compass setting was used to rectangle using only a tape measure. locate points R and S, so . If one pair of SOLUTION: opposite sides of a quadrilateral is both parallel and Sample answer: If the diagonals of a parallelogram congruent, then the quadrilateral is a parallelogram. are congruent, then it is a rectangle. He should A parallelogram with right angles is a rectangle. measure the diagonals of the end zone and both

Thus, PRSQ is a rectangle. pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then ANSWER: the end zone is a rectangle.

ANSWER: Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that Sample answer: He should measure the diagonals of are perpendicular to the same line are parallel, so the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are . The same compass setting was used to congruent, then the end zone is a rectangle. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and ALGEBRA Quadrilateral WXYZ is a rectangle. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 WZ .

38. SPORTS The end zone of a football field is 160 feet Therefore, YW = 5. wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. ANSWER: Explain how Kyle can confirm that the end zone is 5 the regulation size and be sure that it is also a rectangle using only a tape measure. 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: Sample answer: If the diagonals of a parallelogram SOLUTION: are congruent, then it is a rectangle. He should The diagonals of a rectangle are congruent to each measure the diagonals of the end zone and both other. So, WY = XZ = 2c. All four angles of a pairs of opposite sides. If both pairs of opposite sides rectangle are right angles. So, is a right are congruent, then the quadrilateral is a triangle. By the Pythagorean Theorem, XZ2 = XY2 + parallelogram. If the diagonals are congruent, then 2 the end zone is a rectangle. ZY .

ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. Therefore, WY = 2(5) = 10. ALGEBRA Quadrilateral WXYZ is a rectangle. ANSWER: 10

41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko 39. If XW = 3, WZ = 4, and XZ = b, find YW. be sure that the sign is a rectangle? Explain your reasoning. SOLUTION: The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 WZ .

SOLUTION: Therefore, YW = 5. Both pairs of opposite sides are congruent, so the ANSWER: sign is a parallelogram, but no measure is given that 5 can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. angles could be proven to be right angles, then we could prove that the sign was a rectangle. SOLUTION: The diagonals of a rectangle are congruent to each ANSWER: other. So, WY = XZ = 2c. All four angles of a No; sample answer: Both pairs of opposite sides are rectangle are right angles. So, is a right congruent, so the sign is a parallelogram, but no 2 2 measure is given that can be used to prove that it is a triangle. By the Pythagorean Theorem, XZ = XY + 2 rectangle. ZY . PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a Therefore, WY = 2(5) = 10. coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b ANSWER: units. Then the rest of the vertices are A(0, b), B(a, 10 b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what 41. SIGNS The sign below is in the foyer of Nyoko’s you need to prove. Here, you are given ABCD is a school. Based on the dimensions given, can Nyoko rectangle and you need to prove . Use the be sure that the sign is a rectangle? Explain your properties that you have learned about rectangles to reasoning. walk through the proof.

Given: ABCD is a rectangle. Prove:

SOLUTION:

Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that Proof: can be used to prove that it is a rectangle. If the Use the Distance Formula to find the lengths of the diagonals could be proven to be congruent, or if the diagonals. angles could be proven to be right angles, then we could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are The diagonals, have the same length, so congruent, so the sign is a parallelogram, but no they are congruent. measure is given that can be used to prove that it is a rectangle. ANSWER: Given: ABCD is a rectangle. PROOF Write a coordinate proof of each Prove: statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what Proof: you need to prove. Here, you are given ABCD is a Use the Distance Formula to find rectangle and you need to prove . Use the and . properties that you have learned about rectangles to have the same length, so they are walk through the proof. congruent.

Given: ABCD is a rectangle. 43. If the diagonals of a parallelogram are congruent, Prove: then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given Proof: and what you need to prove. Here, you are given Use the Distance Formula to find the lengths of the ABCD and and you need to prove that diagonals. ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

The diagonals, have the same length, so Given: ABCD and they are congruent. Prove: ABCD is a rectangle.

ANSWER: Given: ABCD is a rectangle. Prove:

Proof:

Proof: AC = BD. So, Use the Distance Formula to find and . have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. Because A and B are different points, SOLUTION: a ≠ 0. Then b = 0. The slope of is undefined Begin by positioning parallelogram ABCD on a and the slope of . Thus, . ∠DAB is coordinate plane. Place vertex A at the origin. Let a right angle and ABCD is a rectangle. the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + ANSWER: a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given Given: ABCD and and what you need to prove. Here, you are given Prove: ABCD is a rectangle. ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Prove: ABCD is a rectangle. Proof:

But AC = BD. So,

Proof:

AC = BD. So, Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. Because A and B are different points, a. GEOMETRIC Draw three parallelograms, each a ≠ 0. Then b = 0. The slope of is undefined with all four sides congruent. Label one and the slope of . Thus, . ∠DAB is parallelogram ABCD, one MNOP, and one WXYZ. a right angle and ABCD is a rectangle. Draw the two diagonals of each parallelogram and label the intersections R. ANSWER: b. TABULAR Use a protractor to measure the Given: ABCD and appropriate angles and complete the table below. Prove: ABCD is a rectangle.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

Proof:

But AC = BD. So,

b. Use a protractor to measure each angle listed in the table. Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined

and the slope of . Thus, . ∠DAB is c. Sample answer: Each of the angles listed in the a right angle and ABCD is a rectangle.. table are right angles. These angles were each formed by diagonals. The diagonals of a 44. MULTIPLE REPRESENTATIONS In the parallelogram with four congruent sides are problem, you will explore properties of other special perpendicular. parallelograms. a. GEOMETRIC Draw three parallelograms, each ANSWER: with all four sides congruent. Label one a. Sample answer: parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular. 45. CHALLENGE In rectangle ABCD, Find the values of x and y.

b. Use a protractor to measure each angle listed in the table.

ANSWER:

a. Sample answer: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So,

The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem, b.

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular. ANSWER: CHALLENGE 45. In rectangle ABCD, x = 6, y = −10 Find the values of x and y. 46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

SOLUTION: All four angles of a rectangle are right angles. So,

In order for the quadrilateral to be a rectangle, one of The diagonals of a rectangle are congruent and the angles in the congruent triangles has to be a right bisect each other. So, is an isosceles triangle angle. with So,

The sum of the angles of a triangle is 180. So, Tamika is correct. So, ANSWER: By the Vertical Angle Theorem, Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle. ANSWER: 47. REASONING In the diagram at the right, lines n, p, x = 6, y = −10 q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting 46. CCSS CRITIQUE Parker says that any two lines? congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

So, Tamika is correct. Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 ANSWER: rectangles are formed by the intersecting lines. Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of ANSWER: the angles are formed by a single vertex of a triangle. 6 In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right 48. OPEN ENDED Write the equations of four lines angle. having intersections that form the vertices of a rectangle. Verify your answer using coordinate 47. REASONING In the diagram at the right, lines n, p, geometry. q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting SOLUTION: lines? Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

x = 0, x = 6, y = 0, y = 4; SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

Then the rectangles formed are ABXW, ACYW, Since is horizontal and is vertical, the lines BCYX, BDZX, CDZY, and ADZW. Therefore, 6 are perpendicular and the measure of the angle they rectangles are formed by the intersecting lines. form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by ANSWER: definition, the parallelogram is a rectangle. 6 ANSWER: 48. OPEN ENDED Write the equations of four lines Sample answer: x = 0, x = 6, y = 0, having intersections that form the vertices of a y = 4; rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 The length of is 6 − 0 or 6 units and the length of sides. is 6 – 0 or 6 units. The slope of AB is 0 and the

slope of DC is 0. Since a pair of sides of the

x = 0, x = 6, y = 0, y = 4; quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not The length of is 6 − 0 or 6 units and the length of rectangles. is 6 – 0 or 6 units. The slope of AB is 0 and the SOLUTION: slope of DC is 0. Since a pair of sides of the Sample answer: A parallelogram is a quadrilateral in quadrilateral is both parallel and congruent, by which each pair of opposite sides are parallel. By Theorem 6.12, the quadrilateral is a parallelogram. definition of rectangle, all rectangles have both pairs Since is horizontal and is vertical, the lines of opposite sides parallel. So rectangles are always are perpendicular and the measure of the angle they parallelograms. form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by Parallelograms with right angles are rectangles, so definition, the parallelogram is a rectangle. some parallelograms are rectangles, but others with non-right angles are not. ANSWER: Sample answer: x = 0, x = 6, y = 0, ANSWER: y = 4; Sample answer: All rectangles are parallelograms because, by definition, both pairs of opposite sides are parallel. Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram FGHJ a rectangle?

rectangles. then it is a rectangle. Since FGHJ is a parallelogram, SOLUTION: the diagonals bisect each other. So, if it is a Sample answer: A parallelogram is a quadrilateral in rectangle, then will be an isosceles triangle which each pair of opposite sides are parallel. By with That is, 3x + y = 13. definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always Add the two equations to eliminate the x-term and parallelograms. solve for y. –3x + 5y = 11 Parallelograms with right angles are rectangles, so 3x + y = 13 some parallelograms are rectangles, but others with 6y = 24 Add to eliminate x-term. non-right angles are not. y = 4 Divide each side by 6.

ANSWER: Use the value of y to find the value of x. Sample answer: All rectangles are parallelograms 3x + 4 = 13 Substitute. because, by definition, both pairs of opposite sides 3x = 9 Subtract 4 from each side. are parallel. Parallelograms with right angles are x = 3 Divide each side by 3. rectangles, so some parallelograms are rectangles, but others with non-right angles are not. So, x = 3, y = 4, the correct choice is A.

So, FJ = GH. given by P = 2(l + w). The perimeter is given to be 8 –3x + 5y = 11 FJ = GH ft. The width is r and the length is r + 10. So, 2r + 2

(r + 10) = 80. If the diagonals of a parallelogram are congruent, Therefore, the correct choice is J. then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a ANSWER: rectangle, then will be an isosceles triangle J with That is, 3x + y = 13. 52. SHORT RESPONSE What is the measure of Add the two equations to eliminate the x-term and ? solve for y. –3x + 5y = 11 3x + y = 13 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6.

Use the value of y to find the value of x. 3x + 4 = 13 Substitute. SOLUTION: 3x = 9 Subtract 4 from each side. The diagonals of a rectangle are congruent and x = 3 Divide each side by 3. bisect each other. So, is an isosceles triangle with So, x = 3, y = 4, the correct choice is A. So, By the Exterior Angle Theorem, ANSWER: A ANSWER: 51. ALGEBRA A rectangular playground is surrounded 112 by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following 53. SAT/ACT If p is odd, which of the following must equations could be used to find r, the shorter side of also be odd? the playground? A 2p F 10r + r = 80 B 2p + 2 G 4r + 10 = 80 H r(r + 10) = 80 C J 2(r + 10) + 2r = 80 D SOLUTION: E p + 2 The formula to find the perimeter of a rectangle is SOLUTION: given by P = 2(l + w). The perimeter is given to be 8 Analyze each answer choice to determine which ft. The width is r and the length is r + 10. So, 2r + 2 expression is odd when p is odd. (r + 10) = 80. Choice A 2p Therefore, the correct choice is J. This is a multiple of 2 so it is even. ANSWER: Choice B 2p + 2 J This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even. 52. SHORT RESPONSE What is the measure of ? Choice C If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd SOLUTION: number. This is odd. The diagonals of a rectangle are congruent and The correct choice is E. bisect each other. So, is an isosceles triangle ANSWER: with E So,

By the Exterior Angle Theorem, ALGEBRA Find x and y so that the quadrilateral is a parallelogram. ANSWER: 112

53. SAT/ACT If p is odd, which of the following must also be odd? 54. A 2p SOLUTION: B 2p + 2 The opposite sides of a parallelogram are congruent.

C D

E p + 2 In a parallelogram, consecutive interior angles are SOLUTION: supplementary. Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even. ANSWER: x = 2, y = 41 Choice C If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd 55. number. This is odd. SOLUTION: The correct choice is E. The opposite sides of a parallelogram are congruent.

ANSWER: E

ALGEBRA Find x and y so that the The opposite sides of a parallelogram are parallel to quadrilateral is a parallelogram. each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

54.

SOLUTION: ANSWER: The opposite sides of a parallelogram are congruent. x = 8, y = 22

In a parallelogram, consecutive interior angles are supplementary. 56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 ANSWER: y = 7 x = 2, y = 41 Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: 55. x = 2, y = 7 SOLUTION: 57. COORDINATE GEOMETRY Find the The opposite sides of a parallelogram are congruent. coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: The opposite sides of a parallelogram are parallel to The diagonals of a parallelogram bisect each other, each other. So, the angles with the measures (3y + so the intersection is the midpoint of either diagonal. 3)° and (4y – 19)° are alternate interior angles and Find the midpoint of or . hence they are equal.

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5). ANSWER: ANSWER: x = 8, y = 22 (2.5, 0.5)

Refer to the figure.

56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 58. If , name two congruent angles. y = 7 Use the value of y in the second equation and solve SOLUTION: for x. Since is an isosceles triangle. So, 7 + 3 = 2x + 6 4 = 2x 2 = x ANSWER: ANSWER: x = 2, y = 7 59. If , name two congruent segments. 57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of SOLUTION: parallelogram ABCD with vertices A(1, 3), B(6, 2), C In a triangle with two congruent angles, the sides (4, –2), and D(–1, –1). opposite to the congruent angles are also congruent.

SOLUTION: So, . The diagonals of a parallelogram bisect each other, ANSWER: so the intersection is the midpoint of either diagonal. Find the midpoint of or . 60. If , name two congruent segments. SOLUTION: Therefore, the coordinates of the intersection of the In a triangle with two congruent angles, the sides diagonals are (2.5, 0.5). opposite to the congruent angles are also congruent. ANSWER: So, . (2.5, 0.5) ANSWER: Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. SOLUTION: FARMING An X-brace on a rectangular barn The diagonals of a rectangle are congruent and door is both decorative and functional. It helps bisect each other. So, is an isosceles triangle. to prevent the door from warping over time. Then, If feet, PS = 7 feet, and By the Exterior Angle Theorem,

, find each measure.

Therefore,

ANSWER: 33.5

4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, 1. QR By the Vertical Angle Theorem, SOLUTION: The opposite sides of a rectangle are parallel and Therefore, congruent. Therefore, QR = PS = 7 ft.

ANSWER: ANSWER: 7 ft 56.5 ALGEBRA Quadrilateral DEFG is a rectangle. 2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

ANSWER:

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 3. 11 SOLUTION: The diagonals of a rectangle are congruent and 6. If , bisect each other. So, is an isosceles triangle. find . Then, SOLUTION: By the Exterior Angle Theorem, All four angles of a rectangle are right angles. So,

Therefore,

ANSWER: 33.5

4. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 51 bisect each other. So, is an isosceles triangle. Then, 7. PROOF If ABDE is a rectangle and , By the Vertical Angle Theorem, prove that .

Therefore,

ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and . You need to prove . Use the 5. If FD = 3x – 7 and EG = x + 5, find EG. properties that you have learned about rectangles to SOLUTION: walk through the proof.

The diagonals of a rectangle are congruent to each other. So, FD = EG. Given: ABDE is a rectangle; Prove:

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 11 Statements(Reasons)

6. If , 1. ABDE is a rectangle; . (Given) find . 2. ABDE is a parallelogram. (Def. of SOLUTION: rectangle) All four angles of a rectangle are right angles. So, 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; ANSWER: Prove: 51

7. PROOF If ABDE is a rectangle and , prove that .

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of

rectangle) SOLUTION: 3. (Opp. sides of a .) You need to walk through the proof step by step. 4. are right angles.(Def. of rectangle) Look over what you are given and what you need to prove. You are given ABDE is a rectangle and 5. (All rt .)

. You need to prove . Use the 6. (SAS) properties that you have learned about rectangles to 7. (CPCTC) walk through the proof. COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine Given: ABDE is a rectangle; whether the figure is a rectangle. Justify your Prove: answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) Use the slope formula to find the slope of the sides of 5. (All rt .) the quadrilateral. 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

Statements(Reasons) No; slope of , slope of , slope of 1. ABDE is a rectangle; .

(Given) , and slope of . Slope of 2. ABDE is a parallelogram. (Def. of rectangle) slope of , and slope of slope of

3. (Opp. sides of a .) , so WXYZ is not a parallelogram. Therefore, 4. are right angles.(Def. of rectangle) WXYZ is not a rectangle. 5. (All rt .) 6. (SAS) 7. (CPCTC)

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. so ABCD is a parallelogram. ANSWER: A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the No; slope of , slope of , slope of lengths of the diagonals.

, and slope of . Slope of So the diagonals are congruent. Thus, ABCD is a slope of , and slope of slope of rectangle. , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION: FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral. 10. BC

SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

so ABCD is a ANSWER: parallelogram. 2 ft A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the 11. DB lengths of the diagonals. SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean So the diagonals are congruent. Thus, ABCD is a 2 2 2 rectangle. Theorem, DB = BC + DC . BC = AD = 2 ANSWER: DC = AB = 6 Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: 6.3 ft

12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, By the Vertical Angle Theorem, AD = 2 feet, and , find each measure. ANSWER: 50

13. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 10. BC Then, All four angles of a rectangle are right angles. SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: ANSWER: 2 ft 25

11. DB CCSS REGULARITY Quadrilateral WXYZ is a rectangle. SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6 14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. ANSWER: 2x + 3 = x + 4 6.3 ft x = 1 Use the value of x to find WX.

12. WX = 1 + 4 = 5. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 5 bisect each other. So, is an isosceles triangle. Then, 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP. By the Vertical Angle Theorem,

ANSWER: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 50

13. SOLUTION: The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles 43 triangle. Then, 16. If , find

All four angles of a rectangle are right angles. .

SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 25 CCSS REGULARITY Quadrilateral WXYZ is a rectangle.

ANSWER: 39

17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. 14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and The opposite sides of a rectangle are parallel and bisect each other. So, ZP = PY. congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 Use the value of x to find WX. Then ZP = 4(7) – 9 = 19. WX = 1 + 4 = 5. Therefore, ZX = 2(ZP) = 38. ANSWER: 6-4 Rectangles ANSWER: 5 38

15. If PY = 3x – 5 and WP = 2x + 11, find ZP. 18. If , find . SOLUTION: The diagonals of a rectangle bisect each other. So, SOLUTION: PY = WP. All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

ANSWER: 48 ANSWER: 43 19. If , find . 16. If , find . SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: is a right triangle. Then, All four angles of a rectangle are right angles. So,

Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, ANSWER: = = 46. 39 ANSWER: 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. 46 SOLUTION: PROOF Write a two-column proof. The diagonals of a rectangle are congruent and 20. Given: ABCD is a rectangle. bisect each other. So, ZP = PY. Prove:

Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38. SOLUTION: ANSWER: You need to walk through the proof step by step. 38 Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. 18. If , find You need to prove . Use the . properties that you have learned about rectangles to SOLUTION: walk through the proof. All four angles of a rectangle are right angles. So,

Given: ABCD is a rectangle. eSolutions Manual - Powered by Cognero Prove: Page 5

Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) ANSWER: 4. (Refl. Prop.) 48 5. (Diagonals of a rectangle are .) 19. If , find 6. (SSS) . ANSWER: SOLUTION: Proof: All four angles of a rectangle are right angles. So, Statements (Reasons)

is a right triangle. Then, 1. ABCD is a rectangle. (Given)

2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS) Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, 21. Given: QTVW is a rectangle. = = 46.

ANSWER: 46

PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. Prove: Prove: SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and . You need to prove . Use SOLUTION: the properties that you have learned about rectangles to walk through the proof. You need to walk through the proof step by step.

Look over what you are given and what you need to Given: QTVW is a rectangle. prove. Here, you are given ABCD is a rectangle.

You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove:

Proof: Prove: Statements (Reasons) Proof: 1. ABCD is a rectangle. (Given) Statements (Reasons) 2. ABCD is a parallelogram. (Def. of rectangle) 1. QTVW is a rectangle; .(Given) 3. (Opp. sides of a .) 2. QTVW is a parallelogram. (Def. of rectangle)

4. (Refl. Prop.) 3. (Opp sides of a .) 5. (Diagonals of a rectangle are .) 4. are right angles.(Def. of rectangle) 6. (SSS) 5. (All rt .) 6. QR = ST (Def. of segs.) ANSWER: 7. (Refl. Prop.) Proof:

Statements (Reasons) 8. RS = RS (Def. of segs.) 1. ABCD is a rectangle. (Given) 9. QR + RS = RS + ST (Add. prop.) 2. ABCD is a parallelogram. (Def. of rectangle) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 3. (Opp. sides of a .) 12. (Def. of segs.) 4. (Refl. Prop.) 13. (SAS) 5. (Diagonals of a rectangle are .) 6. (SSS) ANSWER: Proof: 21. Given: QTVW is a rectangle. Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .)

Prove: 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) SOLUTION: 8. RS = RS (Def. of segs.) You need to walk through the proof step by step. 9. QR + RS = RS + ST (Add. prop.) Look over what you are given and what you need to 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) prove. Here, you are given QTVW is a rectangle and 11. QS = RT (Subst.) . You need to prove . Use 12. (Def. of segs.) the properties that you have learned about rectangles 13. (SAS) to walk through the proof.

COORDINATE GEOMETRY Graph each Given: QTVW is a rectangle. quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION:

Prove: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) Use the slope formula to find the slope of the sides of 8. RS = RS (Def. of segs.) the quadrilateral. 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) The slopes of each pair of opposite sides are equal. 2. QTVW is a parallelogram. (Def. of rectangle) So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. 3. (Opp sides of a .) The products of the slopes of the adjacent sides are 4. are right angles.(Def. of rectangle) –1. So, any two adjacent sides are perpendicular to 5. (All rt .) each other. That is, all the four angles are right 6. QR = ST (Def. of segs.) angles. Therefore, WXYZ is a rectangle. 7. (Refl. Prop.) ANSWER: 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) Yes; slope of slope of , slope 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) of slope of . So WXYZ is a 11. QS = RT (Subst.) parallelogram. The product of the slopes of

12. (Def. of segs.) consecutive sides is –1, so the consecutive sides are 13. (SAS) perpendicular and form right angles. Thus, WXYZ is a rectangle. COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION:

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

ANSWER: parallelogram. A parallelogram is a rectangle if the diagonals are Yes; slope of slope of , slope congruent. Use the Distance formula to find the lengths of the diagonals. of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are So the diagonals are not congruent. Thus, JKLM is perpendicular and form right angles. Thus, WXYZ is a not a rectangle. rectangle. ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals. so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are So the diagonals are not congruent. Thus, JKLM is congruent. Use the Distance Formula to find the not a rectangle. lengths of the diagonals.

ANSWER: No; so JKLM is a So the diagonals are congruent. Thus, QRST is a parallelogram; so rectangle. the diagonals are not congruent. Thus, JKLM is not a rectangle. ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula

SOLUTION: 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

First, use the slope formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle. The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. ANSWER: Therefore, the quadrilateral GHJK is a Yes; so QRST is a parallelogram. parallelogram. so the diagonals are None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. congruent. QRST is a rectangle. Therefore, GHJK is not a rectangle.

ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

First, use the slope formula to find the lengths of the sides of the quadrilateral.

26. SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: The slopes of each pair of opposite sides are equal. 50 So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a 27. parallelogram. None of the adjacent sides have slopes whose SOLUTION: product is –1. So, the angles are not right angles. The measures of angles 2 and 3 are equal as they Therefore, GHJK is not a rectangle. are alternate interior angles. Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 3, 7 No; slope of slope of and slope of and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. slope of . So, GHJK is a parallelogram. The product of the slopes of ANSWER: consecutive sides ≠ –1, so the consecutive sides are 40 not perpendicular. Thus, GHJK is not a rectangle. 28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

Quadrilateral ABCD is a rectangle. Find each measure if . The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 26. and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. SOLUTION: Therefore, m∠5 = 180 – (50 + 50) = 80. All four angles of a rectangle are right angles. So, ANSWER: 80 30. ANSWER: SOLUTION: 50 All four angles of a rectangle are right angles. So,

27. SOLUTION: The measures of angles 1 and 4 are equal as they The measures of angles 2 and 3 are equal as they are alternate interior angles. are alternate interior angles.

Since the diagonals of a rectangle are congruent and Therefore, m∠4 = m∠1 = 50. bisect each other, the triangle with the angles 3, 7 Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 8 is an isosceles triangle. So, m∠7 = m∠3. and 6 is an isosceles triangle. Therefore, m∠6 = Therefore, m∠7 = m∠2 = 40. m∠4 = 50. ANSWER: ANSWER: 40 50 28. 31. SOLUTION: SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m 3 = m 2 = 40. ∠ ∠ Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 3, 7

40 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. 29. Therefore, m∠8 = 180 – (40 + 40) = 100. SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 100

are alternate interior angles. which means that the angles formed by the shelves Therefore, m∠4 = m∠1 = 50. and the vertical supports are all right angles? Explain Since the diagonals of a rectangle are congruent and your reasoning. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: 80

30. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find the lengths of the metal supports. The measures of angles 1 and 4 are equal as they Let x be the length of each support. Express 3 feet are alternate interior angles. as 36 inches. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = m∠4 = 50. Therefore, the metal supports should each be 39 ANSWER: inches long or 3 feet 3 inches long. 50 ANSWER: 31. 3 ft 3 in.; Sample answer: In order for the bookshelf SOLUTION: to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the The measures of angles 2 and 3 are equal as they bookshelves and the distance between them, I used are alternate interior angles. the Pythagorean Theorem to find the lengths of the Since the diagonals of a rectangle are congruent and metal supports. bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. PROOF Write a two-column proof. The sum of the three angles of a triangle is 180. 33. Theorem 6.13 Therefore, m∠8 = 180 – (40 + 40) = 100. SOLUTION: ANSWER: You need to walk through the proof step by step. 100 Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with 32. CCSS MODELING Jody is building a new diagonals . You need to prove bookshelf using wood and metal supports like the one shown. To what length should she cut the metal . Use the properties that you have learned supports in order for the bookshelf to be square, about rectangles to walk through the proof. which means that the angles formed by the shelves and the vertical supports are all right angles? Explain Given: WXYZ is a rectangle with diagonals your reasoning. . Prove:

Proof: 1. WXYZ is a rectangle with diagonals . SOLUTION: (Given) In order for the bookshelf to be square, the lengths of 2. (Opp. sides of a .) all of the metal supports should be equal. Since we know the length of the bookshelves and the distance 3. (Refl. Prop.) between them, use the Pythagorean Theorem to find 4. are right angles. (Def. of the lengths of the metal supports. rectangle) Let x be the length of each support. Express 3 feet 5. (All right .) as 36 inches. 6. (SAS)

7. (CPCTC)

ANSWER: Given: WXYZ is a rectangle with diagonals

Therefore, the metal supports should each be 39 . inches long or 3 feet 3 inches long. Prove: ANSWER: 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the

bookshelves and the distance between them, I used Proof: the Pythagorean Theorem to find the lengths of the metal supports. 1. WXYZ is a rectangle with diagonals . (Given) PROOF Write a two-column proof. 2. (Opp. sides of a .) 33. Theorem 6.13 3. (Refl. Prop.) SOLUTION: 4. are right angles. (Def. of You need to walk through the proof step by step. rectangle) Look over what you are given and what you need to 5. (All right .) prove. Here, you are given WXYZ is a rectangle with 6. (SAS) diagonals . You need to prove 7. (CPCTC) . Use the properties that you have learned about rectangles to walk through the proof. 34. Theorem 6.14 Given: WXYZ is a rectangle with diagonals SOLUTION: . You need to walk through the proof step by step. Prove: Look over what you are given and what you need to prove. Here, you are given . You need to prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk through the proof. Proof: 1. WXYZ is a rectangle with diagonals . Given: (Given) Prove: WXYZ is a rectangle. 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle)

5. (All right .) Proof:

6. (SAS) 1. (Given)

7. (CPCTC) 2. (SSS)

ANSWER: 3. (CPCTC) Given: WXYZ is a rectangle with diagonals 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. . sides are , then quad. is .)

Prove: 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) Proof: 9. are right angles. (If a has 1 1. WXYZ is a rectangle with diagonals . rt. , it has 4 rt. .) (Given) 10. WXYZ is a rectangle. (Def. of rectangle) 2. (Opp. sides of a .) ANSWER: 3. (Refl. Prop.) Given: 4. are right angles. (Def. of Prove: WXYZ is a rectangle. rectangle) 5. (All right .) 6. (SAS) 7. (CPCTC)

34. Theorem 6.14 Proof: 1. (Given) SOLUTION: 2. (SSS) You need to walk through the proof step by step. Look over what you are given and what you need to 3. (CPCTC) prove. Here, you are given 4. (Def. of ) . You need to 5. WXYZ is a parallelogram. (If both pairs of opp. prove that WXYZ is a rectangle.. Use the properties sides are , then quad. is .) that you have learned about rectangles to walk 6. are supplementary. (Cons. through the proof. are suppl.) 7. (Def of suppl.)

Given: 8. are right angles. (If Prove: WXYZ is a rectangle. 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each Proof: statement. 1. (Given) 35. If a parallelogram has one right angle, then it is a rectangle. 2. (SSS) 3. (CPCTC) SOLUTION: 4. (Def. of ) You need to walk through the proof step by step. 5. WXYZ is a parallelogram. (If both pairs of opp. Look over what you are given and what you need to sides are , then quad. is .) prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. 6. are supplementary. (Cons. Let ABCD be a parallelogram with one right angle. are suppl.) Then use the properties of parallelograms to walk 7. (Def of suppl.) through the proof. 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

ANSWER: ABCD is a parallelogram, and is a right angle. Given: Since ABCD is a parallelogram and has one right Prove: WXYZ is a rectangle. angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle.

ANSWER:

Proof: 1. (Given) 2. (SSS) 3. (CPCTC) ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right 4. (Def. of ) angle, then it has four right angles. So by the 5. WXYZ is a parallelogram. (If both pairs of opp. definition of a rectangle, ABCD is a rectangle. sides are , then quad. is .) 6. are supplementary. (Cons. 36. If a quadrilateral has four right angles, then it is a are suppl.) rectangle. 7. (Def of suppl.) SOLUTION: 8. are right angles. (If You need to walk through the proof step by step.

2 and suppl., each is a rt. .) Look over what you are given and what you need to 9. are right angles. (If a has 1 prove. Here, you are given a quadrilateral with four rt. , it has 4 rt. .) right angles. You need to prove that it is a rectangle. 10. WXYZ is a rectangle. (Def. of rectangle) Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and PROOF Write a paragraph proof of each rectangles to walk through the proof. statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one ABCD is a quadrilateral with four right angles. right angle. You need to prove that it is a rectangle. ABCD is a parallelogram because both pairs of Let ABCD be a parallelogram with one right angle. opposite angles are congruent. By definition of a Then use the properties of parallelograms to walk rectangle, ABCD is a rectangle. through the proof. ANSWER:

ABCD is a parallelogram, and is a right angle. ABCD is a quadrilateral with four right angles. Since ABCD is a parallelogram and has one right ABCD is a parallelogram because both pairs of angle, then it has four right angles. So by the opposite angles are congruent. By definition of a definition of a rectangle, ABCD is a rectangle. rectangle, ABCD is a rectangle. ANSWER: 37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction.

SOLUTION: ABCD is a parallelogram, and is a right angle. Sample answer: Since ABCD is a parallelogram and has one right Step 1: Graph a line and place points P and Q on the angle, then it has four right angles. So by the line. definition of a rectangle, ABCD is a rectangle. Step 2: Place the compass on point P. Using the 36. If a quadrilateral has four right angles, then it is a same compass setting, draw an arc to the left and rectangle. right of P that intersects . SOLUTION: You need to walk through the proof step by step. Step 3: Open the compass to a setting greater than Look over what you are given and what you need to the distance from P to either point of intersection. prove. Here, you are given a quadrilateral with four Place the compass on each point of intersection and right angles. You need to prove that it is a rectangle. draw arcs that intersect above . Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and Step 4: Draw line a through point P and the rectangles to walk through the proof. intersection of the arcs drawn in step 3.

Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

ABCD is a quadrilateral with four right angles. Step 7: Using any compass setting, put the compass ABCD is a parallelogram because both pairs of at P and draw an arc above it that intersects line a. opposite angles are congruent. By definition of a Label the point of intersection R. rectangle, ABCD is a rectangle. Step 8: Using the same compass setting, put the ANSWER: compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw .

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. Since and , the measure of SOLUTION: angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Sample answer: Step 1: Graph a line and place points P and Q on the . The same compass setting was used to line. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and Step 2: Place the compass on point P. Using the congruent, then the quadrilateral is a parallelogram. same compass setting, draw an arc to the left and A parallelogram with right angles is a rectangle. right of P that intersects . Thus, PRSQ is a rectangle.

Step 3: Open the compass to a setting greater than ANSWER: the distance from P to either point of intersection. Place the compass on each point of intersection and draw arcs that intersect above . Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3. . The same compass setting was used to locate points R and S, so . If one pair of Step 5: Repeat steps 2 and 3 using point Q. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 6: Draw line b through point Q and the A parallelogram with right angles is a rectangle. intersection of the arcs drawn in step 5. Thus, PRSQ is a rectangle.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw .

38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both Since and , the measure of pairs of opposite sides. If both pairs of opposite sides angles P and Q is 90 degrees. Lines that are are congruent, then the quadrilateral is a perpendicular to the same line are parallel, so parallelogram. If the diagonals are congruent, then . The same compass setting was used to the end zone is a rectangle. locate points R and S, so . If one pair of ANSWER: opposite sides of a quadrilateral is both parallel and Sample answer: He should measure the diagonals of congruent, then the quadrilateral is a parallelogram. the end zone and both pairs of opposite sides. If the A parallelogram with right angles is a rectangle. diagonals and both pairs of opposite sides are

Thus, PRSQ is a rectangle. congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle. ANSWER:

Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that

are perpendicular to the same line are parallel, so 39. If XW = 3, WZ = 4, and XZ = b, find YW. . The same compass setting was used to SOLUTION: locate points R and S, so . If one pair of The diagonals of a rectangle are congruent to each opposite sides of a quadrilateral is both parallel and other. So, YW = XZ = b. All four angles of a congruent, then the quadrilateral is a parallelogram. rectangle are right angles. So, is a right A parallelogram with right angles is a rectangle. 2 2 Thus, PRSQ is a rectangle. triangle. By the Pythagorean Theorem, XZ = XW + 2 WZ .

Therefore, YW = 5.

ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: 38. SPORTS The end zone of a football field is 160 feet The diagonals of a rectangle are congruent to each wide and 30 feet long. Kyle is responsible for other. So, WY = XZ = 2c. All four angles of a painting the field. He has finished the end zone. rectangle are right angles. So, is a right Explain how Kyle can confirm that the end zone is triangle. By the Pythagorean Theorem, XZ2 = XY2 + the regulation size and be sure that it is also a 2 ZY . rectangle using only a tape measure.

SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides

are congruent, then the quadrilateral is a Therefore, WY = 2(5) = 10. parallelogram. If the diagonals are congruent, then ANSWER: the end zone is a rectangle. 10 ANSWER: 41. SIGNS The sign below is in the foyer of Nyoko’s Sample answer: He should measure the diagonals of school. Based on the dimensions given, can Nyoko the end zone and both pairs of opposite sides. If the be sure that the sign is a rectangle? Explain your diagonals and both pairs of opposite sides are reasoning. congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

ANSWER: Therefore, YW = 5. No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no ANSWER: measure is given that can be used to prove that it is a 5 rectangle.

PROOF Write a coordinate proof of each 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. statement. SOLUTION: 42. The diagonals of a rectangle are congruent. The diagonals of a rectangle are congruent to each SOLUTION: other. So, WY = XZ = 2c. All four angles of a Begin by positioning quadrilateral ABCD on a rectangle are right angles. So, is a right coordinate plane. Place vertex D at the origin. Let 2 2 triangle. By the Pythagorean Theorem, XZ = XY + the length of the bases be a units and the height be b 2 ZY . units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to walk through the proof. Therefore, WY = 2(5) = 10. Given: ABCD is a rectangle. ANSWER: Prove: 10

41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your reasoning.

Proof: Use the Distance Formula to find the lengths of the diagonals.

SOLUTION: The diagonals, have the same length, so Both pairs of opposite sides are congruent, so the they are congruent. sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the ANSWER: diagonals could be proven to be congruent, or if the Given: ABCD is a rectangle. angles could be proven to be right angles, then we Prove: could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

coordinate plane. Place vertex D at the origin. Let congruent. the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, 43. If the diagonals of a parallelogram are congruent, b), and C(a, 0). You need to walk through the proof then it is a rectangle. step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a SOLUTION: Begin by positioning parallelogram ABCD on a rectangle and you need to prove . Use the coordinate plane. Place vertex A at the origin. Let properties that you have learned about rectangles to the length of the bases be a units and the height be c walk through the proof. units. Then the rest of the vertices are B(a, 0), C(b +

a, c), and D(b, c). You need to walk through the Given: ABCD is a rectangle. proof step by step. Look over what you are given Prove: and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Prove: ABCD is a rectangle. Proof: Use the Distance Formula to find the lengths of the diagonals.

The diagonals, have the same length, so Proof: they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove: AC = BD. So,

Proof: Because A and B are different points, Use the Distance Formula to find a ≠ 0. Then b = 0. The slope of is undefined and . and the slope of . Thus, . ∠DAB is have the same length, so they are a right angle and ABCD is a rectangle. congruent. ANSWER: Given: ABCD and 43. If the diagonals of a parallelogram are congruent, then it is a rectangle. Prove: ABCD is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given Proof: and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you

have learned about rectangles to walk through the But AC = BD. So, proof.

Given: ABCD and Prove: ABCD is a rectangle.

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined

Proof: and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.. 44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. AC = BD. So, a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined c. VERBAL Make a conjecture about the diagonals and the slope of . Thus, . ∠DAB is of a parallelogram with four congruent sides. a right angle and ABCD is a rectangle. SOLUTION: ANSWER: a. Sample answer: Given: ABCD and Prove: ABCD is a rectangle.

Proof:

b. Use a protractor to measure each angle listed in

But AC = BD. So, the table.

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

Because A and B are different points, ANSWER: a ≠ 0. Then b = 0. The slope of is undefined a. Sample answer: and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

c. VERBAL Make a conjecture about the diagonals c. Sample answer: The diagonals of a parallelogram of a parallelogram with four congruent sides. with four congruent sides are perpendicular. SOLUTION: 45. CHALLENGE In rectangle ABCD, a. Sample answer: Find the values of x and y.

SOLUTION:

b. Use a protractor to measure each angle listed in All four angles of a rectangle are right angles. So, the table.

c. Sample answer: Each of the angles listed in the The diagonals of a rectangle are congruent and table are right angles. These angles were each bisect each other. So, is an isosceles triangle formed by diagonals. The diagonals of a with parallelogram with four congruent sides are So, perpendicular.

ANSWER: a. Sample answer: The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

ANSWER: x = 6, y = −10

b. 46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is c. Sample answer: The diagonals of a parallelogram either of them correct? Explain your reasoning. with four congruent sides are perpendicular. SOLUTION: 45. CHALLENGE In rectangle ABCD, When two congruent triangles are arranged to form a Find the quadrilateral, two of the angles are formed by a values of x and y. single vertex of a triangle.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and So, Tamika is correct. bisect each other. So, is an isosceles triangle with ANSWER: So, Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle. The sum of the angles of a triangle is 180. So, 47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. By the Vertical Angle Theorem, How many rectangles are formed by the intersecting lines?

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a SOLUTION: rectangle. Tamika says that only two congruent right Let the lines n, p, q, and r intersect the line l at the triangles can be arranged to make a rectangle. Is points A, B, C, and D respectively. Similarly, let the either of them correct? Explain your reasoning. lines intersect the line m at the points W, X, Y, and Z. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle. Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a So, Tamika is correct. rectangle. Verify your answer using coordinate geometry. ANSWER: Tamika; Sample answer: When two congruent SOLUTION: triangles are arranged to form a quadrilateral, two of Sample answer: the angles are formed by a single vertex of a triangle. A rectangle has 4 right angles and each pair of In order for the quadrilateral to be a rectangle, one of opposite sides is parallel and congruent. Lines that the angles in the congruent triangles has to be a right run parallel to the x-axis will be parallel to each other angle. and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the 47. REASONING In the diagram at the right, lines n, p, rectangle. The lines y = 4 and x = 6 form the other 2 q,and r are parallel and lines l and m are parallel. sides. How many rectangles are formed by the intersecting lines? x = 0, x = 6, y = 0, y = 4;

SOLUTION: Let the lines n, p, q, and r intersect the line l at the The length of is 6 − 0 or 6 units and the length of points A, B, C, and D respectively. Similarly, let the is 6 – 0 or 6 units. The slope of AB is 0 and the lines intersect the line m at the points W, X, Y, and Z. slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4; Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. The length of is 6 − 0 or 6 units and the length of SOLUTION: is 6 – 0 or 6 units. The slope of AB is 0 and the Sample answer: slope of DC is 0. Since a pair of sides of the A rectangle has 4 right angles and each pair of quadrilateral is both parallel and congruent, by opposite sides is parallel and congruent. Lines that Theorem 6.12, the quadrilateral is a parallelogram. run parallel to the x-axis will be parallel to each other Since is horizontal and is vertical, the lines and perpendicular to lines parallel to the y-axis. are perpendicular and the measure of the angle they Selecting y = 0 and x = 0 will form 2 sides of the form is 90. By Theorem 6.6, if a parallelogram has rectangle. The lines y = 4 and x = 6 form the other 2 one right angle, it has four right angles. Therefore, by sides. definition, the parallelogram is a rectangle.

x = 0, x = 6, y = 0, y = 4; 49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not rectangles. SOLUTION: Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms.

The length of is 6 − 0 or 6 units and the length of Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with is 6 – 0 or 6 units. The slope of AB is 0 and the non-right angles are not. slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by ANSWER: Theorem 6.12, the quadrilateral is a parallelogram. Sample answer: All rectangles are parallelograms Since is horizontal and is vertical, the lines because, by definition, both pairs of opposite sides are perpendicular and the measure of the angle they are parallel. Parallelograms with right angles are form is 90. By Theorem 6.6, if a parallelogram has rectangles, so some parallelograms are rectangles, one right angle, it has four right angles. Therefore, by but others with non-right angles are not. definition, the parallelogram is a rectangle. 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = ANSWER: 13, what values of x and y make parallelogram Sample answer: x = 0, x = 6, y = 0, FGHJ a rectangle? y = 4;

A x = 3, y = 4 B x = 4, y = 3 C x = 7, y = 8 D x = 8, y = 7 SOLUTION: The length of is 6 − 0 or 6 units and the length of The opposite sides of a parallelogram are congruent. is 6 – 0 or 6 units. The slope of AB is 0 and the So, FJ = GH. slope of DC is 0. Since a pair of sides of the –3x + 5y = 11 FJ = GH quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, Since is horizontal and is vertical, the lines the diagonals bisect each other. So, if it is a are perpendicular and the measure of the angle they rectangle, then will be an isosceles triangle form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by with That is, 3x + y = 13. definition, the parallelogram is a rectangle. Add the two equations to eliminate the x-term and 49. WRITING IN MATH Explain why all rectangles solve for y. are parallelograms, but all parallelograms are not –3x + 5y = 11 rectangles. 3x + y = 13 6y = 24 Add to eliminate x-term. SOLUTION: y = 4 Divide each side by 6. Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By Use the value of y to find the value of x. definition of rectangle, all rectangles have both pairs 3x + 4 = 13 Substitute. of opposite sides parallel. So rectangles are always 3x = 9 Subtract 4 from each side. parallelograms. x = 3 Divide each side by 3.

Parallelograms with right angles are rectangles, so So, x = 3, y = 4, the correct choice is A. some parallelograms are rectangles, but others with non-right angles are not. ANSWER: A ANSWER: Sample answer: All rectangles are parallelograms 51. ALGEBRA A rectangular playground is surrounded because, by definition, both pairs of opposite sides by an 80-foot fence. One side of the playground is 10 are parallel. Parallelograms with right angles are feet longer than the other. Which of the following rectangles, so some parallelograms are rectangles, equations could be used to find r, the shorter side of but others with non-right angles are not. the playground? F 10r + r = 80 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram G 4r + 10 = 80 FGHJ a rectangle? H r(r + 10) = 80 J 2(r + 10) + 2r = 80 SOLUTION: The formula to find the perimeter of a rectangle is given by P = 2(l + w). The perimeter is given to be 8 A x = 3, y = 4 ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. B x = 4, y = 3 Therefore, the correct choice is J. C x = 7, y = 8 D x = 8, y = 7 ANSWER: SOLUTION: J The opposite sides of a parallelogram are congruent. 52. SHORT RESPONSE What is the measure of So, FJ = GH. ? –3x + 5y = 11 FJ = GH

If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a rectangle, then will be an isosceles triangle with That is, 3x + y = 13. SOLUTION: Add the two equations to eliminate the x-term and The diagonals of a rectangle are congruent and solve for y. bisect each other. So, is an isosceles triangle –3x + 5y = 11 with

3x + y = 13 So,

6y = 24 Add to eliminate x-term. By the Exterior Angle Theorem, y = 4 Divide each side by 6.

Use the value of y to find the value of x. ANSWER: 3x + 4 = 13 Substitute. 112 3x = 9 Subtract 4 from each side. x = 3 Divide each side by 3. 53. SAT/ACT If p is odd, which of the following must also be odd? So, x = 3, y = 4, the correct choice is A. A 2p ANSWER: B 2p + 2

A C

51. ALGEBRA A rectangular playground is surrounded D by an 80-foot fence. One side of the playground is 10 E p + 2 feet longer than the other. Which of the following equations could be used to find r, the shorter side of SOLUTION: the playground? Analyze each answer choice to determine which F 10r + r = 80 expression is odd when p is odd. Choice A 2p G 4r + 10 = 80 This is a multiple of 2 so it is even. H r(r + 10) = 80 Choice B 2p + 2 J 2(r + 10) + 2r = 80 This can be rewritten as 2(p + 1) which is also a SOLUTION: multiple of 2 so it is even. The formula to find the perimeter of a rectangle is given by P = 2(l + w). The perimeter is given to be 8 Choice C ft. The width is r and the length is r + 10. So, 2r + 2 If p is odd , is not a whole number.

(r + 10) = 80. Choice D 2p – 2

Therefore, the correct choice is J. This can be rewritten as 2(p - 1) which is also a ANSWER: multiple of 2 so it is even. J Choice E p + 2 Adding 2 to an odd number results in an odd 52. SHORT RESPONSE What is the measure of number. This is odd. ? The correct choice is E. ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with 54. So, By the Exterior Angle Theorem, SOLUTION: The opposite sides of a parallelogram are congruent.

ANSWER: 112 In a parallelogram, consecutive interior angles are 53. SAT/ACT If p is odd, which of the following must supplementary. also be odd? A 2p B 2p + 2

C D E p + 2 ANSWER:

SOLUTION: x = 2, y = 41 Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a 55. multiple of 2 so it is even. SOLUTION: Choice C The opposite sides of a parallelogram are congruent. If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. The opposite sides of a parallelogram are parallel to Choice E p + 2 each other. So, the angles with the measures (3y + Adding 2 to an odd number results in an odd 3)° and (4y – 19)° are alternate interior angles and number. This is odd. hence they are equal. The correct choice is E.

ANSWER: E ANSWER: ALGEBRA Find x and y so that the

quadrilateral is a parallelogram. x = 8, y = 22

54. 56. SOLUTION: SOLUTION: The opposite sides of a parallelogram are congruent. The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 In a parallelogram, consecutive interior angles are y = 7 supplementary. Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: x = 2, y = 7 ANSWER: x = 2, y = 41 57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. 55. Find the midpoint of or . SOLUTION: The opposite sides of a parallelogram are congruent.

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

The opposite sides of a parallelogram are parallel to ANSWER: each other. So, the angles with the measures (3y + (2.5, 0.5) 3)° and (4y – 19)° are alternate interior angles and hence they are equal. Refer to the figure.

ANSWER: x = 8, y = 22

58. If , name two congruent angles. SOLUTION: 56. Since is an isosceles triangle. So, SOLUTION: The diagonals of a parallelogram bisect each other. ANSWER: So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 59. If , name two congruent segments. Use the value of y in the second equation and solve for x. SOLUTION: 7 + 3 = 2x + 6 In a triangle with two congruent angles, the sides 4 = 2x opposite to the congruent angles are also congruent. 2 = x So, .

ANSWER: ANSWER: x = 2, y = 7 57. COORDINATE GEOMETRY Find the

coordinates of the intersection of the diagonals of 60. If , name two congruent segments. parallelogram ABCD with vertices A(1, 3), B(6, 2), C SOLUTION: (4, –2), and D(–1, –1). In a triangle with two congruent angles, the sides SOLUTION: opposite to the congruent angles are also congruent. The diagonals of a parallelogram bisect each other, So, . so the intersection is the midpoint of either diagonal. ANSWER: Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem, FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. Therefore,

Therefore,

1. QR ANSWER: SOLUTION: 56.5 The opposite sides of a rectangle are parallel and ALGEBRA Quadrilateral DEFG is a rectangle. congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: SOLUTION: The diagonals of a rectangle bisect each other. So, The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: ANSWER: 11

Therefore,

ANSWER: ANSWER: 51 33.5 PROOF 4. 7. If ABDE is a rectangle and , prove that . SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Given: ABDE is a rectangle;

5. If FD = 3x – 7 and EG = x + 5, find EG. Prove: SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. Statements(Reasons) EG = 6 + 5 = 11 1. ABDE is a rectangle; . (Given) ANSWER: 2. ABDE is a parallelogram. (Def. of 11 rectangle) 3. (Opp. sides of a .) 6. If , 4. are right angles.(Def. of rectangle) find . 5. (All rt .) SOLUTION: 6. (SAS) All four angles of a rectangle are right angles. So, 7. (CPCTC)

ANSWER:

Given: ABDE is a rectangle; Prove:

ANSWER: 51

7. PROOF If ABDE is a rectangle and , Statements(Reasons)

Statements(Reasons) 1. ABDE is a rectangle; . (Given) Use the slope formula to find the slope of the sides of 2. ABDE is a parallelogram. (Def. of the quadrilateral. rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER:

Given: ABDE is a rectangle; There are no parallel sides, so WXYZ is not a Prove: parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of

Use the slope formula to find the slope of the sides of the quadrilateral. First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the

There are no parallel sides, so WXYZ is not a lengths of the diagonals. parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER: So the diagonals are congruent. Thus, ABCD is a

No; slope of , slope of , slope of rectangle. ANSWER: , and slope of . Slope of Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a slope of , and slope of slope of parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

10. BC SOLUTION: The opposite sides of a rectangle are parallel and

First, Use the Distance formula to find the lengths of congruent. Therefore, BC = AD = 2 ft.

the sides of the quadrilateral. ANSWER:

2 ft

11. DB so ABCD is a SOLUTION: parallelogram. All four angles of a rectangle are right angles. So, A parallelogram is a rectangle if the diagonals are is a right triangle. By the Pythagorean 2 2 2 congruent. Use the Distance formula to find the Theorem, DB = BC + DC . lengths of the diagonals. BC = AD = 2

DC = AB = 6

So the diagonals are congruent. Thus, ABCD is a rectangle.

By the Vertical Angle Theorem,

10. BC ANSWER: SOLUTION: 25 The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. CCSS REGULARITY Quadrilateral WXYZ is a rectangle. ANSWER: 2 ft

11. DB SOLUTION:

BC = AD = 2 congruent. So, ZY = WX. DC = AB = 6 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5.

ANSWER: ANSWER: 6.3 ft 5 12. 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles The diagonals of a rectangle bisect each other. So,

triangle. PY = WP.

Then,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. By the Vertical Angle Theorem,

ANSWER: 50 ANSWER: 43 13. 16. If , find SOLUTION:

The diagonals of a rectangle are congruent and . bisect each other. So, is an isosceles SOLUTION: triangle. All four angles of a rectangle are right angles. So, Then, All four angles of a rectangle are right angles.

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: Then ZP = 4(7) – 9 = 19. The opposite sides of a rectangle are parallel and Therefore, ZX = 2(ZP) = 38. congruent. So, ZY = WX. 2x + 3 = x + 4 ANSWER: x = 1 38 Use the value of x to find WX. WX = 1 + 4 = 5. 18. If , find ANSWER: . 5 SOLUTION: All four angles of a rectangle are right angles. So, 15. If PY = 3x – 5 and WP = 2x + 11, find ZP.

SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP.

The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 48

19. If , find . SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 43 is a right triangle. Then,

16. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Therefore, Since is a

transversal of parallel sides and , alternate

interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: 46

ANSWER: PROOF Write a two-column proof. Given: 39 20. ABCD is a rectangle. Prove: 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY.

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to Then ZP = 4(7) – 9 = 19. prove. Here, you are given ABCD is a rectangle. Therefore, ZX = 2(ZP) = 38. You need to prove . Use the ANSWER: properties that you have learned about rectangles to walk through the proof. 38

18. If , find Given: ABCD is a rectangle. Prove: . Proof: SOLUTION: Statements (Reasons) All four angles of a rectangle are right angles. So, 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle)

3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS)

ANSWER: ANSWER: 48 Proof: Statements (Reasons) 19. If , find 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) . 3. (Opp. sides of a .) SOLUTION: 4. (Refl. Prop.) All four angles of a rectangle are right angles. So, 5. (Diagonals of a rectangle are .) is a right triangle. Then, 6. (SSS)

21. Given: QTVW is a rectangle.

Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46. Prove: 6-4 RectanglesANSWER: 46 SOLUTION: You need to walk through the proof step by step. PROOF Write a two-column proof. Look over what you are given and what you need to 20. Given: ABCD is a rectangle. prove. Here, you are given QTVW is a rectangle and Prove: . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: QTVW is a rectangle.

1. ABCD is a rectangle. (Given) 4. are right angles.(Def. of rectangle) 2. ABCD is a parallelogram. (Def. of rectangle) 5. (All rt .) 3. (Opp. sides of a .) 6. QR = ST (Def. of segs.)

4. (Refl. Prop.) 7. (Refl. Prop.)

5. (Diagonals of a rectangle are .) 8. RS = RS (Def. of segs.)

6. (SSS) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) ANSWER: 11. QS = RT (Subst.) Proof: 12. (Def. of segs.) Statements (Reasons) 13. (SAS) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) ANSWER: 3. (Opp. sides of a .) Proof: Statements (Reasons) 4. (Refl. Prop.) 1. QTVW is a rectangle; .(Given) 5. (Diagonals of a rectangle are .) 2. QTVW is a parallelogram. (Def. of rectangle) 6. (SSS) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 21. Given: QTVW is a rectangle. 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) Prove: 11. QS = RT (Subst.) SOLUTION: 12. (Def. of segs.) You need to walk through the proof step by step. 13. (SAS) Look over what you are given and what you need to eSolutionsprove.Manual Here,- Powered you areby givenCognero QTVW is a rectangle and COORDINATE GEOMETRY Graph each Page 6 quadrilateral with the given vertices. Determine . You need to prove . Use whether the figure is a rectangle. Justify your the properties that you have learned about rectangles answer using the indicated formula.

to walk through the proof. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula

Given: QTVW is a rectangle. SOLUTION:

Prove: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) Use the slope formula to find the slope of the sides of 4. are right angles.(Def. of rectangle) the quadrilateral.

2. QTVW is a parallelogram. (Def. of rectangle) angles. Therefore, WXYZ is a rectangle. 3. (Opp sides of a .) ANSWER: 4. are right angles.(Def. of rectangle) Yes; slope of slope of , slope 5. (All rt .) 6. QR = ST (Def. of segs.) of slope of . So WXYZ is a 7. (Refl. Prop.) parallelogram. The product of the slopes of 8. RS = RS (Def. of segs.) consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a 9. QR + RS = RS + ST (Add. prop.) rectangle. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

of slope of . So WXYZ is a ANSWER: parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are No; so JKLM is a perpendicular and form right angles. Thus, WXYZ is a parallelogram; so rectangle. the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance

Formula 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION: SOLUTION:

First, use the Distance Formula to find the lengths of First, Use the Distance formula to find the lengths of the sides of the quadrilateral. the sides of the quadrilateral.

so QRST is a so JKLM is a parallelogram.

parallelogram. A parallelogram is a rectangle if the diagonals are A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the congruent. Use the Distance formula to find the lengths of the diagonals.

lengths of the diagonals.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula

SOLUTION: 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, use the slope formula to find the lengths of the First, use the Distance Formula to find the lengths of sides of the quadrilateral. the sides of the quadrilateral.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

26. SOLUTION: First, use the slope formula to find the lengths of the All four angles of a rectangle are right angles. So, sides of the quadrilateral.

ANSWER: 50

29. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they are alternate interior angles. Quadrilateral ABCD is a rectangle. Find each Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and measure if . bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER:

26. 80 SOLUTION: 30. All four angles of a rectangle are right angles. So, SOLUTION:

All four angles of a rectangle are right angles. So,

Therefore, m∠7 = m∠2 = 40. 31. ANSWER: SOLUTION: 40 The measures of angles 2 and 3 are equal as they are alternate interior angles. 28. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 The measures of angles 2 and 3 are equal as they and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. are alternate interior angles. The sum of the three angles of a triangle is 180.

Therefore, m∠3 = m∠2 = 40. Therefore, m∠8 = 180 – (40 + 40) = 100. ANSWER: ANSWER: 40 100

29. 32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one SOLUTION: shown. To what length should she cut the metal All four angles of a rectangle are right angles. So, supports in order for the bookshelf to be square, which means that the angles formed by the shelves and the vertical supports are all right angles? Explain your reasoning. The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. SOLUTION: ANSWER: In order for the bookshelf to be square, the lengths of 80 all of the metal supports should be equal. Since we know the length of the bookshelves and the distance

30. between them, use the Pythagorean Theorem to find SOLUTION: the lengths of the metal supports. All four angles of a rectangle are right angles. So, Let x be the length of each support. Express 3 feet as 36 inches.

The measures of angles 1 and 4 are equal as they are alternate interior angles.

Therefore, m∠4 = m∠1 = 50. Therefore, the metal supports should each be 39 Since the diagonals of a rectangle are congruent and inches long or 3 feet 3 inches long. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = ANSWER: m∠4 = 50. 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports ANSWER: should be equal. Since I knew the length of the 50 bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the 31. metal supports. SOLUTION: PROOF Write a two-column proof. The measures of angles 2 and 3 are equal as they 33. Theorem 6.13 are alternate interior angles. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 You need to walk through the proof step by step. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. Look over what you are given and what you need to The sum of the three angles of a triangle is 180. prove. Here, you are given WXYZ is a rectangle with Therefore, m∠8 = 180 – (40 + 40) = 100. diagonals . You need to prove ANSWER: . Use the properties that you have learned 100 about rectangles to walk through the proof.

Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of SOLUTION: rectangle) In order for the bookshelf to be square, the lengths of 5. (All right .) all of the metal supports should be equal. Since we 6. (SAS) know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find 7. (CPCTC) the lengths of the metal supports. ANSWER: Let x be the length of each support. Express 3 feet as 36 inches. Given: WXYZ is a rectangle with diagonals . Prove:

Therefore, the metal supports should each be 39 inches long or 3 feet 3 inches long. Proof: ANSWER: 3 ft 3 in.; Sample answer: In order for the bookshelf 1. WXYZ is a rectangle with diagonals . to be square, the lengths of all of the metal supports (Given) should be equal. Since I knew the length of the 2. (Opp. sides of a .) bookshelves and the distance between them, I used 3. (Refl. Prop.) the Pythagorean Theorem to find the lengths of the metal supports. 4. are right angles. (Def. of rectangle) PROOF Write a two-column proof. 5. (All right .) 33. Theorem 6.13 6. (SAS) SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. Look over what you are given and what you need to 34. Theorem 6.14 prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove SOLUTION: . Use the properties that you have learned You need to walk through the proof step by step. Look over what you are given and what you need to about rectangles to walk through the proof. prove. Here, you are given

Given: WXYZ is a rectangle with diagonals . You need to prove that WXYZ is a rectangle.. Use the properties . that you have learned about rectangles to walk Prove: through the proof.

Given: Prove: WXYZ is a rectangle.

Proof: 1. WXYZ is a rectangle with diagonals . (Given)

. 8. are right angles. (If Prove: 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

34. Theorem 6.14 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. SOLUTION: sides are , then quad. is .) You need to walk through the proof step by step. 6. are supplementary. (Cons. Look over what you are given and what you need to are suppl.) prove. Here, you are given 7. (Def of suppl.) . You need to 8. are right angles. (If prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk 2 and suppl., each is a rt. .) through the proof. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) Given: 10. WXYZ is a rectangle. (Def. of rectangle) Prove: WXYZ is a rectangle. PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION:

Proof: You need to walk through the proof step by step. Look over what you are given and what you need to 1. (Given) prove. Here, you are given a parallelogram with one 2. (SSS) right angle. You need to prove that it is a rectangle. 3. (CPCTC) Let ABCD be a parallelogram with one right angle. 4. (Def. of ) Then use the properties of parallelograms to walk 5. WXYZ is a parallelogram. (If both pairs of opp. through the proof. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) ABCD is a parallelogram, and is a right angle. 9. are right angles. (If a has 1 Since ABCD is a parallelogram and has one right rt. , it has 4 rt. .) angle, then it has four right angles. So by the 10. WXYZ is a rectangle. (Def. of rectangle) definition of a rectangle, ABCD is a rectangle.

ANSWER: ANSWER: Given: Prove: WXYZ is a rectangle.

sides are , then quad. is .) You need to walk through the proof step by step. 6. are supplementary. (Cons. Look over what you are given and what you need to are suppl.) prove. Here, you are given a quadrilateral with four 7. (Def of suppl.) right angles. You need to prove that it is a rectangle. 8. are right angles. (If Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and 2 and suppl., each is a rt. .) rectangles to walk through the proof. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. ABCD is a quadrilateral with four right angles. SOLUTION: ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a You need to walk through the proof step by step. rectangle, ABCD is a rectangle. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one ANSWER: right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

Step 2: Place the compass on point P. Using the ABCD is a parallelogram, and is a right angle. same compass setting, draw an arc to the left and Since ABCD is a parallelogram and has one right

angle, then it has four right angles. So by the right of P that intersects . definition of a rectangle, ABCD is a rectangle. Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. 36. If a quadrilateral has four right angles, then it is a Place the compass on each point of intersection and rectangle. draw arcs that intersect above . SOLUTION: You need to walk through the proof step by step. Step 4: Draw line a through point P and the Look over what you are given and what you need to intersection of the arcs drawn in step 3. prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Step 5: Repeat steps 2 and 3 using point Q. Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and Step 6: rectangles to walk through the proof. Draw line b through point Q and the

intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Since and , the measure of 37. CONSTRUCTION Construct a rectangle using the angles P and Q is 90 degrees. Lines that are construction for congruent segments and the perpendicular to the same line are parallel, so construction for a line perpendicular to another line through a point on the line. Justify each step of the . The same compass setting was used to construction. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and SOLUTION: congruent, then the quadrilateral is a parallelogram. Sample answer: A parallelogram with right angles is a rectangle. Step 1: Graph a line and place points P and Q on the Thus, PRSQ is a rectangle. line.

congruent, then the quadrilateral is a parallelogram. Step 4: Draw line a through point P and the A parallelogram with right angles is a rectangle. intersection of the arcs drawn in step 3. Thus, PRSQ is a rectangle.

Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S. 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Step 9: Draw . Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then the end zone is a rectangle. Since and , the measure of ANSWER: angles P and Q is 90 degrees. Lines that are Sample answer: He should measure the diagonals of perpendicular to the same line are parallel, so the end zone and both pairs of opposite sides. If the . The same compass setting was used to diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and ALGEBRA Quadrilateral WXYZ is a rectangle. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

ANSWER: 39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: Sample answer: Since and , the The diagonals of a rectangle are congruent to each measure of angles P and Q is 90 degrees. Lines that other. So, YW = XZ = b. All four angles of a are perpendicular to the same line are parallel, so rectangle are right angles. So, is a right . The same compass setting was used to triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 locate points R and S, so . If one pair of WZ . opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Therefore, YW = 5. ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: The diagonals of a rectangle are congruent to each other. So, WY = XZ = 2c. All four angles of a rectangle are right angles. So, is a right 2 2 triangle. By the Pythagorean Theorem, XZ = XY + 2 ZY . SPORTS T 38. he end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Therefore, WY = 2(5) = 10. Sample answer: If the diagonals of a parallelogram ANSWER: are congruent, then it is a rectangle. He should 10 measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides 41. SIGNS The sign below is in the foyer of Nyoko’s are congruent, then the quadrilateral is a school. Based on the dimensions given, can Nyoko parallelogram. If the diagonals are congruent, then be sure that the sign is a rectangle? Explain your

the end zone is a rectangle. reasoning. ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

SOLUTION:

Both pairs of opposite sides are congruent, so the

39. If XW = 3, WZ = 4, and XZ = b, find YW. sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the SOLUTION: diagonals could be proven to be congruent, or if the The diagonals of a rectangle are congruent to each angles could be proven to be right angles, then we other. So, YW = XZ = b. All four angles of a could prove that the sign was a rectangle. rectangle are right angles. So, is a right 2 2 ANSWER: triangle. By the Pythagorean Theorem, XZ = XW + No; sample answer: Both pairs of opposite sides are 2 WZ . congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

Therefore, YW = 5. PROOF Write a coordinate proof of each statement. ANSWER: 42. The diagonals of a rectangle are congruent. 5 SOLUTION: Begin by positioning quadrilateral ABCD on a 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. coordinate plane. Place vertex D at the origin. Let SOLUTION: the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, The diagonals of a rectangle are congruent to each b), and C(a, 0). You need to walk through the proof other. So, WY = XZ = 2c. All four angles of a step by step. Look over what you are given and what rectangle are right angles. So, is a right you need to prove. Here, you are given ABCD is a 2 2 triangle. By the Pythagorean Theorem, XZ = XY + rectangle and you need to prove . Use the 2 ZY . properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove:

Therefore, WY = 2(5) = 10.

ANSWER: 10

41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko Proof: be sure that the sign is a rectangle? Explain your Use the Distance Formula to find the lengths of the reasoning. diagonals.

The diagonals, have the same length, so they are congruent.

ANSWER:

No; sample answer: Both pairs of opposite sides are Proof: congruent, so the sign is a parallelogram, but no Use the Distance Formula to find measure is given that can be used to prove that it is a rectangle. and . have the same length, so they are PROOF Write a coordinate proof of each congruent. statement. 42. The diagonals of a rectangle are congruent. 43. If the diagonals of a parallelogram are congruent, SOLUTION: then it is a rectangle. Begin by positioning quadrilateral ABCD on a SOLUTION: coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b Begin by positioning parallelogram ABCD on a units. Then the rest of the vertices are A(0, b), B(a, coordinate plane. Place vertex A at the origin. Let b), and C(a, 0). You need to walk through the proof the length of the bases be a units and the height be c step by step. Look over what you are given and what units. Then the rest of the vertices are B(a, 0), C(b + you need to prove. Here, you are given ABCD is a a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given rectangle and you need to prove . Use the and what you need to prove. Here, you are given properties that you have learned about rectangles to walk through the proof. ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you Given: ABCD is a rectangle. have learned about rectangles to walk through the proof. Prove:

Given: ABCD and Prove: ABCD is a rectangle.

Proof: Use the Distance Formula to find the lengths of the diagonals. Proof:

The diagonals, have the same length, so AC = BD. So, they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.

Proof: ANSWER: Use the Distance Formula to find Given: ABCD and and . Prove: ABCD is a rectangle. have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a Proof: coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the But AC = BD. So, proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special Proof: parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. AC = BD. So, b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

ANSWER: Given: ABCD and Prove: ABCD is a rectangle.

b. Use a protractor to measure each angle listed in the table. Proof:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each

with all four sides congruent. Label one b. parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the c. Sample answer: The diagonals of a parallelogram appropriate angles and complete the table below. with four congruent sides are perpendicular. 45. CHALLENGE In rectangle ABCD, Find the c. VERBAL Make a conjecture about the diagonals values of x and y. of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and b. Use a protractor to measure each angle listed in bisect each other. So, is an isosceles triangle

the table. with So,

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular. The sum of the angles of a triangle is 180. So, ANSWER: By the Vertical Angle Theorem, a. Sample answer:

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. b. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a c. Sample answer: The diagonals of a parallelogram single vertex of a triangle. with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines?

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the SOLUTION: quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the Since is horizontal and is vertical, the lines lines intersect the line m at the points W, X, Y, and Z. are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

Sample answer: A parallelogram is a quadrilateral in

x = 0, x = 6, y = 0, y = 4; which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms.

Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

ANSWER:

Parallelograms with right angles are rectangles, so 51. ALGEBRA A rectangular playground is surrounded some parallelograms are rectangles, but others with by an 80-foot fence. One side of the playground is 10 non-right angles are not. feet longer than the other. Which of the following equations could be used to find r, the shorter side of ANSWER: the playground? Sample answer: All rectangles are parallelograms F 10r + r = 80 because, by definition, both pairs of opposite sides G 4r + 10 = 80 are parallel. Parallelograms with right angles are H r(r + 10) = 80 rectangles, so some parallelograms are rectangles, but others with non-right angles are not. J 2(r + 10) + 2r = 80 SOLUTION: 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram The formula to find the perimeter of a rectangle is FGHJ a rectangle? given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. Therefore, the correct choice is J.

3x + y = 13 112 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6. 53. SAT/ACT If p is odd, which of the following must also be odd? Use the value of y to find the value of x. A 2p 3x + 4 = 13 Substitute. B 2p + 2 3x = 9 Subtract 4 from each side.

x = 3 Divide each side by 3. C

So, x = 3, y = 4, the correct choice is A. D E p + 2 ANSWER: SOLUTION: A Analyze each answer choice to determine which 51. ALGEBRA A rectangular playground is surrounded expression is odd when p is odd. by an 80-foot fence. One side of the playground is 10 Choice A 2p feet longer than the other. Which of the following This is a multiple of 2 so it is even. equations could be used to find r, the shorter side of Choice B 2p + 2 the playground? This can be rewritten as 2(p + 1) which is also a F 10r + r = 80 multiple of 2 so it is even. G 4r + 10 = 80 Choice C H r(r + 10) = 80 J 2(r + 10) + 2r = 80 If p is odd , is not a whole number. Choice D 2p – 2 SOLUTION: This can be rewritten as 2(p - 1) which is also a The formula to find the perimeter of a rectangle is multiple of 2 so it is even. given by P = 2(l + w). The perimeter is given to be 8

ft. The width is r and the length is r + 10. So, 2r + 2 Choice E p + 2 (r + 10) = 80. Adding 2 to an odd number results in an odd Therefore, the correct choice is J. number. This is odd. The correct choice is E. ANSWER: J ANSWER: E 52. SHORT RESPONSE What is the measure of ? ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

ANSWER: 112

53. SAT/ACT If p is odd, which of the following must also be odd? A 2p ANSWER: B 2p + 2

x = 2, y = 41 C D E p + 2 SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. 55. Choice A 2p SOLUTION: This is a multiple of 2 so it is even. The opposite sides of a parallelogram are congruent. Choice B 2 p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). ANSWER: x = 2, y = 41 SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

55. Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5). SOLUTION: The opposite sides of a parallelogram are congruent. ANSWER: (2.5, 0.5) Refer to the figure.

The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

58. If , name two congruent angles. ANSWER: x = 8, y = 22 SOLUTION: Since is an isosceles triangle. So,

ANSWER:

56.

SOLUTION: 59. If , name two congruent segments. The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. SOLUTION: Solve the first equation to find the value of y. In a triangle with two congruent angles, the sides 2y = 14 opposite to the congruent angles are also congruent. y = 7 So, . Use the value of y in the second equation and solve for x. ANSWER: 7 + 3 = 2x + 6 4 = 2x 2 = x 60. If , name two congruent segments. ANSWER: SOLUTION: x = 2, y = 7 In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. 57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of So, . parallelogram ABCD with vertices A(1, 3), B(6, 2), C ANSWER: (4, –2), and D(–1, –1).

SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

, find each measure. Therefore,

ANSWER: 33.5

4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, 1. QR By the Vertical Angle Theorem,

SOLUTION:

The opposite sides of a rectangle are parallel and Therefore, congruent. Therefore, QR = PS = 7 ft. ANSWER: ANSWER: 56.5 7 ft ALGEBRA Quadrilateral DEFG is a rectangle. 2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

ANSWER: Use the value of x to find EG.

EG = 6 + 5 = 11

Therefore,

ANSWER: 33.5

4. ANSWER: SOLUTION: 51 The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 7. PROOF If ABDE is a rectangle and ,

Then, By the Vertical Angle Theorem, prove that .

Therefore,

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 11 Statements(Reasons) 1. ABDE is a rectangle; . 6. If , (Given) find . 2. ABDE is a parallelogram. (Def. of rectangle) SOLUTION:

All four angles of a rectangle are right angles. So, 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; ANSWER: Prove: 51

7. PROOF If ABDE is a rectangle and , prove that .

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) SOLUTION: 3. (Opp. sides of a .) You need to walk through the proof step by step. 4. are right angles.(Def. of rectangle) Look over what you are given and what you need to 5. (All rt .) prove. You are given ABDE is a rectangle and 6. (SAS) . You need to prove . Use the

properties that you have learned about rectangles to 7. (CPCTC) walk through the proof. COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine Given: ABDE is a rectangle; whether the figure is a rectangle. Justify your Prove: answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .)

4. are right angles.(Def. of rectangle) Use the slope formula to find the slope of the sides of 5. (All rt .) the quadrilateral. 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

Statements(Reasons) No; slope of , slope of , slope of 1. ABDE is a rectangle; . (Given) , and slope of . Slope of 2. ABDE is a parallelogram. (Def. of rectangle) slope of , and slope of slope of 3. (Opp. sides of a .) , so WXYZ is not a parallelogram. Therefore, 4. are right angles.(Def. of rectangle) WXYZ is not a rectangle. 5. (All rt .) 6. (SAS) 7. (CPCTC)

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

There are no parallel sides, so WXYZ is not a so ABCD is a

parallelogram. Therefore, WXYZ is not a rectangle. parallelogram. ANSWER: A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the No; slope of , slope of , slope of lengths of the diagonals.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral. 10. BC SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: so ABCD is a parallelogram. 2 ft A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the 11. DB lengths of the diagonals. SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean So the diagonals are congruent. Thus, ABCD is a 2 2 2 rectangle. Theorem, DB = BC + DC . BC = AD = 2 ANSWER: DC = AB = 6 Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. ANSWER: 6.3 ft

12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, By the Vertical Angle Theorem, AD = 2 feet, and , find each measure. ANSWER: 50

13. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

10. BC Then, All four angles of a rectangle are right angles. SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: ANSWER: 2 ft 25

DC = AB = 6 14. If ZY = 2x + 3 and WX = x + 4, find WX.

SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. ANSWER: 2x + 3 = x + 4 6.3 ft x = 1 Use the value of x to find WX. 12. WX = 1 + 4 = 5.

SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 5 bisect each other. So, is an isosceles triangle. 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. Then, SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP. By the Vertical Angle Theorem,

The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 50 13. SOLUTION: The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles 43 triangle. Then, 16. If , find All four angles of a rectangle are right angles. .

SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 25 CCSS REGULARITY Quadrilateral WXYZ is a

rectangle.

ANSWER: 39

ANSWER: ANSWER: 5 38

15. If PY = 3x – 5 and WP = 2x + 11, find ZP. 18. If , find . SOLUTION: The diagonals of a rectangle bisect each other. So, SOLUTION: PY = WP. All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

Therefore, Since is a

transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, ANSWER: = = 46. 39 ANSWER: 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. 46 SOLUTION: PROOF Write a two-column proof. The diagonals of a rectangle are congruent and 20. Given: ABCD is a rectangle. bisect each other. So, ZP = PY. Prove:

All four angles of a rectangle are right angles. So, Given: ABCD is a rectangle.

Prove: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) ANSWER: 4. (Refl. Prop.) 48 5. (Diagonals of a rectangle are .) 6. (SSS) 19. If , find . ANSWER: SOLUTION: Proof: Statements (Reasons) All four angles of a rectangle are right angles. So, 1. ABCD is a rectangle. (Given) is a right triangle. Then, 2. ABCD is a parallelogram. (Def. of rectangle)

3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS) Therefore, Since is a transversal of parallel sides and , alternate 21. Given: QTVW is a rectangle. interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: 46 PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. Prove: Prove: SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and . You need to prove . Use the properties that you have learned about rectangles SOLUTION: to walk through the proof. You need to walk through the proof step by step. Look over what you are given and what you need to Given: QTVW is a rectangle. prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle.

Prove: Prove: Proof: Proof: Statements (Reasons) Statements (Reasons) 1. ABCD is a rectangle. (Given)

2. ABCD is a parallelogram. (Def. of rectangle) 1. QTVW is a rectangle; .(Given)

3. (Opp. sides of a .) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. (Refl. Prop.) 4. are right angles.(Def. of rectangle) 5. (Diagonals of a rectangle are .) 5. (All rt .) 6. (SSS) 6. QR = ST (Def. of segs.) ANSWER: 7. (Refl. Prop.) Proof: 8. RS = RS (Def. of segs.) Statements (Reasons) 9. QR + RS = RS + ST (Add. prop.) 1. ABCD is a rectangle. (Given) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 2. ABCD is a parallelogram. (Def. of rectangle) 11. QS = RT (Subst.) 3. (Opp. sides of a .) 12. (Def. of segs.)

4. (Refl. Prop.) 13. (SAS) 5. (Diagonals of a rectangle are .) 6. (SSS) ANSWER: Proof: Statements (Reasons) Given: 21. QTVW is a rectangle. 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) Prove: 7. (Refl. Prop.) SOLUTION: 8. RS = RS (Def. of segs.) You need to walk through the proof step by step. 9. QR + RS = RS + ST (Add. prop.) Look over what you are given and what you need to 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) prove. Here, you are given QTVW is a rectangle and 11. QS = RT (Subst.) . You need to prove . Use 12. (Def. of segs.) the properties that you have learned about rectangles 13. (SAS) to walk through the proof. COORDINATE GEOMETRY Graph each Given: QTVW is a rectangle. quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION:

9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) The slopes of each pair of opposite sides are equal. 1. QTVW is a rectangle; .(Given) So, the two pairs of opposite sides are parallel.

2. QTVW is a parallelogram. (Def. of rectangle) Therefore, the quadrilateral WXYZ is a parallelogram. 3. (Opp sides of a .) The products of the slopes of the adjacent sides are 4. are right angles.(Def. of rectangle) –1. So, any two adjacent sides are perpendicular to 5. (All rt .) each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle. 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) ANSWER: 8. RS = RS (Def. of segs.) Yes; slope of slope of , slope 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) of slope of . So WXYZ is a 11. QS = RT (Subst.) parallelogram. The product of the slopes of 6-4 Rectangles12. (Def. of segs.) consecutive sides is –1, so the consecutive sides are 13. (SAS) perpendicular and form right angles. Thus, WXYZ is a rectangle. COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION:

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of The slopes of each pair of opposite sides are equal. the sides of the quadrilateral. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle. so JKLM is a parallelogram. ANSWER: A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the Yes; slope of slope of , slope lengths of the diagonals. of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are So the diagonals are not congruent. Thus, JKLM is perpendicular and form right angles. Thus, WXYZ is a not a rectangle. rectangle. ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle. eSolutions Manual - Powered by Cognero Page 7

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

not a rectangle. lengths of the diagonals.

ANSWER: No; so JKLM is a So the diagonals are congruent. Thus, QRST is a parallelogram; so rectangle. the diagonals are not congruent. Thus, JKLM is not a ANSWER: rectangle. Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula

SOLUTION: 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

First, use the slope formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle. The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. ANSWER: Therefore, the quadrilateral GHJK is a Yes; so QRST is a parallelogram. None of the adjacent sides have slopes whose parallelogram. so the diagonals are product is –1. So, the angles are not right angles. congruent. QRST is a rectangle. Therefore, GHJK is not a rectangle.

ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

First, use the slope formula to find the lengths of the sides of the quadrilateral. 26. SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: The slopes of each pair of opposite sides are equal. 50 So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a 27. parallelogram. None of the adjacent sides have slopes whose SOLUTION: product is –1. So, the angles are not right angles. The measures of angles 2 and 3 are equal as they Therefore, GHJK is not a rectangle. are alternate interior angles. Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. No; slope of slope of and slope of Therefore, m∠7 = m∠2 = 40. slope of . So, GHJK is a ANSWER: parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are 40 not perpendicular. Thus, GHJK is not a rectangle. 28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

Quadrilateral ABCD is a rectangle. Find each measure if . The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. 26. The sum of the three angles of a triangle is 180. SOLUTION: Therefore, m∠5 = 180 – (50 + 50) = 80. All four angles of a rectangle are right angles. So, ANSWER: 80

30. ANSWER: SOLUTION: 50 All four angles of a rectangle are right angles. So,

27.

SOLUTION: The measures of angles 1 and 4 are equal as they The measures of angles 2 and 3 are equal as they are alternate interior angles. are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 bisect each other, the triangle with the angles 4, 5 and 8 is an isosceles triangle. So, m∠7 = m∠3. and 6 is an isosceles triangle. Therefore, m∠6 = Therefore, m∠7 = m∠2 = 40. m∠4 = 50. ANSWER: ANSWER: 40 50

29. Therefore, m∠8 = 180 – (40 + 40) = 100. SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 100

32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one shown. To what length should she cut the metal The measures of angles 1 and 4 are equal as they supports in order for the bookshelf to be square, are alternate interior angles. which means that the angles formed by the shelves Therefore, m∠4 = m∠1 = 50. and the vertical supports are all right angles? Explain Since the diagonals of a rectangle are congruent and your reasoning. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: 80

30. SOLUTION: SOLUTION: In order for the bookshelf to be square, the lengths of All four angles of a rectangle are right angles. So, all of the metal supports should be equal. Since we know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find the lengths of the metal supports. The measures of angles 1 and 4 are equal as they Let x be the length of each support. Express 3 feet are alternate interior angles. as 36 inches. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = m∠4 = 50. Therefore, the metal supports should each be 39 ANSWER: inches long or 3 feet 3 inches long. 50 ANSWER: 31. 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports SOLUTION: should be equal. Since I knew the length of the The measures of angles 2 and 3 are equal as they bookshelves and the distance between them, I used are alternate interior angles. the Pythagorean Theorem to find the lengths of the Since the diagonals of a rectangle are congruent and metal supports. bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. PROOF Write a two-column proof. The sum of the three angles of a triangle is 180. 33. Theorem 6.13 Therefore, m 8 = 180 – (40 + 40) = 100. ∠ SOLUTION: ANSWER: You need to walk through the proof step by step. 100 Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with 32. CCSS MODELING Jody is building a new diagonals . You need to prove bookshelf using wood and metal supports like the one . Use the properties that you have learned shown. To what length should she cut the metal about rectangles to walk through the proof. supports in order for the bookshelf to be square, which means that the angles formed by the shelves and the vertical supports are all right angles? Explain Given: WXYZ is a rectangle with diagonals your reasoning. . Prove:

Proof: 1. WXYZ is a rectangle with diagonals . SOLUTION: (Given) In order for the bookshelf to be square, the lengths of 2. (Opp. sides of a .) all of the metal supports should be equal. Since we 3. (Refl. Prop.) know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find 4. are right angles. (Def. of the lengths of the metal supports. rectangle) Let x be the length of each support. Express 3 feet 5. (All right .) as 36 inches. 6. (SAS) 7. (CPCTC)

ANSWER: Given: WXYZ is a rectangle with diagonals . Therefore, the metal supports should each be 39 inches long or 3 feet 3 inches long. Prove:

ANSWER: 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the bookshelves and the distance between them, I used Proof: the Pythagorean Theorem to find the lengths of the 1. WXYZ is a rectangle with diagonals . metal supports. (Given) PROOF Write a two-column proof. 2. (Opp. sides of a .) 33. Theorem 6.13 3. (Refl. Prop.) SOLUTION: 4. are right angles. (Def. of You need to walk through the proof step by step. rectangle) Look over what you are given and what you need to 5. (All right .) prove. Here, you are given WXYZ is a rectangle with 6. (SAS) diagonals . You need to prove 7. (CPCTC) . Use the properties that you have learned about rectangles to walk through the proof. 34. Theorem 6.14

Given: WXYZ is a rectangle with diagonals SOLUTION: . You need to walk through the proof step by step. Look over what you are given and what you need to Prove: prove. Here, you are given . You need to prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk

through the proof. Proof:

1. WXYZ is a rectangle with diagonals . Given: (Given) Prove: WXYZ is a rectangle. 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) 5. (All right .) Proof: 6. (SAS) 1. (Given) 7. (CPCTC) 2. (SSS) 3. (CPCTC) ANSWER: 4. (Def. of ) Given: WXYZ is a rectangle with diagonals 5. WXYZ is a parallelogram. (If both pairs of opp.

. sides are , then quad. is .) Prove: 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If

2 and suppl., each is a rt. .) Proof: 9. are right angles. (If a has 1 1. WXYZ is a rectangle with diagonals . rt. , it has 4 rt. .) (Given) 10. WXYZ is a rectangle. (Def. of rectangle)

2. (Opp. sides of a .) ANSWER:

3. (Refl. Prop.) Given: 4. are right angles. (Def. of Prove: WXYZ is a rectangle. rectangle) 5. (All right .) 6. (SAS) 7. (CPCTC)

Proof: 34. Theorem 6.14 1. (Given) SOLUTION: 2. (SSS) You need to walk through the proof step by step. 3. (CPCTC) Look over what you are given and what you need to prove. Here, you are given 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. . You need to sides are , then quad. is .) prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk 6. are supplementary. (Cons. through the proof. are suppl.) 7. (Def of suppl.) Given: 8. are right angles. (If Prove: WXYZ is a rectangle. 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each Proof: statement. 35. If a parallelogram has one right angle, then it is a 1. (Given) rectangle. 2. (SSS) 3. (CPCTC) SOLUTION: 4. (Def. of ) You need to walk through the proof step by step. Look over what you are given and what you need to 5. WXYZ is a parallelogram. (If both pairs of opp. prove. Here, you are given a parallelogram with one sides are , then quad. is .) right angle. You need to prove that it is a rectangle. 6. are supplementary. (Cons. Let ABCD be a parallelogram with one right angle. are suppl.) Then use the properties of parallelograms to walk 7. (Def of suppl.) through the proof. 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

ANSWER:

Proof: 1. (Given) 2. (SSS) ABCD is a parallelogram, and is a right angle. 3. (CPCTC) Since ABCD is a parallelogram and has one right 4. (Def. of ) angle, then it has four right angles. So by the 5. WXYZ is a parallelogram. (If both pairs of opp. definition of a rectangle, ABCD is a rectangle. sides are , then quad. is .) 6. are supplementary. (Cons. 36. If a quadrilateral has four right angles, then it is a are suppl.) rectangle. 7. (Def of suppl.) SOLUTION: 8. are right angles. (If You need to walk through the proof step by step. 2 and suppl., each is a rt. .) Look over what you are given and what you need to 9. are right angles. (If a has 1 prove. Here, you are given a quadrilateral with four rt. , it has 4 rt. .) right angles. You need to prove that it is a rectangle. 10. WXYZ is a rectangle. (Def. of rectangle) Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and PROOF Write a paragraph proof of each rectangles to walk through the proof. statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one ABCD is a quadrilateral with four right angles. right angle. You need to prove that it is a rectangle. ABCD is a parallelogram because both pairs of Let ABCD be a parallelogram with one right angle. opposite angles are congruent. By definition of a Then use the properties of parallelograms to walk rectangle, ABCD is a rectangle. through the proof. ANSWER:

ANSWER: 37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. SOLUTION:

ABCD is a parallelogram, and is a right angle. Sample answer: Since ABCD is a parallelogram and has one right Step 1: Graph a line and place points P and Q on the angle, then it has four right angles. So by the line. definition of a rectangle, ABCD is a rectangle. Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and 36. If a quadrilateral has four right angles, then it is a rectangle. right of P that intersects .

SOLUTION: Step 3: Open the compass to a setting greater than You need to walk through the proof step by step. the distance from P to either point of intersection. Look over what you are given and what you need to Place the compass on each point of intersection and prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. draw arcs that intersect above . Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and Step 4: Draw line a through point P and the rectangles to walk through the proof. intersection of the arcs drawn in step 3.

Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass

ABCD is a quadrilateral with four right angles. at P and draw an arc above it that intersects line a. ABCD is a parallelogram because both pairs of Label the point of intersection R. opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 8: Using the same compass setting, put the ANSWER: compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw .

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. Since and , the measure of angles P and Q is 90 degrees. Lines that are SOLUTION: perpendicular to the same line are parallel, so Sample answer: Step 1: Graph a line and place points P and Q on the . The same compass setting was used to line. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and Step 2: Place the compass on point P. Using the congruent, then the quadrilateral is a parallelogram. same compass setting, draw an arc to the left and A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

right of P that intersects .

Step 3: Open the compass to a setting greater than ANSWER: the distance from P to either point of intersection. Place the compass on each point of intersection and Sample answer: Since and , the

draw arcs that intersect above . measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3. . The same compass setting was used to locate points R and S, so . If one pair of Step 5: Repeat steps 2 and 3 using point Q. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 6: Draw line b through point Q and the A parallelogram with right angles is a rectangle. intersection of the arcs drawn in step 5. Thus, PRSQ is a rectangle.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw .

measure the diagonals of the end zone and both Since and , the measure of pairs of opposite sides. If both pairs of opposite sides angles P and Q is 90 degrees. Lines that are are congruent, then the quadrilateral is a perpendicular to the same line are parallel, so parallelogram. If the diagonals are congruent, then . The same compass setting was used to the end zone is a rectangle. locate points R and S, so . If one pair of ANSWER: opposite sides of a quadrilateral is both parallel and Sample answer: He should measure the diagonals of congruent, then the quadrilateral is a parallelogram. the end zone and both pairs of opposite sides. If the A parallelogram with right angles is a rectangle. diagonals and both pairs of opposite sides are Thus, PRSQ is a rectangle. congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle. ANSWER:

Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so 39. If XW = 3, WZ = 4, and XZ = b, find YW. . The same compass setting was used to SOLUTION: locate points R and S, so . If one pair of The diagonals of a rectangle are congruent to each opposite sides of a quadrilateral is both parallel and other. So, YW = XZ = b. All four angles of a congruent, then the quadrilateral is a parallelogram. rectangle are right angles. So, is a right A parallelogram with right angles is a rectangle. triangle. By the Pythagorean Theorem, XZ2 = XW 2 + Thus, PRSQ is a rectangle. 2 WZ .

Therefore, YW = 5.

ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: The diagonals of a rectangle are congruent to each 38. SPORTS The end zone of a football field is 160 feet other. So, WY = XZ = 2c. All four angles of a wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. rectangle are right angles. So, is a right 2 2 Explain how Kyle can confirm that the end zone is triangle. By the Pythagorean Theorem, XZ = XY + 2 the regulation size and be sure that it is also a ZY . rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides Therefore, WY = 2(5) = 10. are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then ANSWER: the end zone is a rectangle. 10 ANSWER: 41. SIGNS The sign below is in the foyer of Nyoko’s Sample answer: He should measure the diagonals of school. Based on the dimensions given, can Nyoko the end zone and both pairs of opposite sides. If the be sure that the sign is a rectangle? Explain your diagonals and both pairs of opposite sides are reasoning. congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that rectangle are right angles. So, is a right 2 2 can be used to prove that it is a rectangle. If the triangle. By the Pythagorean Theorem, XZ = XW + diagonals could be proven to be congruent, or if the 2 WZ . angles could be proven to be right angles, then we could prove that the sign was a rectangle. ANSWER: Therefore, YW = 5. No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no ANSWER: measure is given that can be used to prove that it is a 5 rectangle. PROOF Write a coordinate proof of each 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. statement. SOLUTION: 42. The diagonals of a rectangle are congruent. The diagonals of a rectangle are congruent to each SOLUTION: other. So, WY = XZ = 2c. All four angles of a Begin by positioning quadrilateral ABCD on a rectangle are right angles. So, is a right coordinate plane. Place vertex D at the origin. Let triangle. By the Pythagorean Theorem, XZ2 = XY2 + the length of the bases be a units and the height be b 2 units. Then the rest of the vertices are A(0, b), B(a, ZY . b), and C(a, 0). You need to walk through the proof

step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to walk through the proof. Therefore, WY = 2(5) = 10. Given: ABCD is a rectangle. ANSWER: Prove: 10

41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your reasoning.

Proof: Use the Distance Formula to find the lengths of the diagonals.

ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

PROOF Write a coordinate proof of each Proof: statement. Use the Distance Formula to find 42. The diagonals of a rectangle are congruent. and . SOLUTION: have the same length, so they are Begin by positioning quadrilateral ABCD on a congruent. coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, 43. If the diagonals of a parallelogram are congruent, b), and C(a, 0). You need to walk through the proof then it is a rectangle. step by step. Look over what you are given and what SOLUTION: you need to prove. Here, you are given ABCD is a Begin by positioning parallelogram ABCD on a rectangle and you need to prove . Use the coordinate plane. Place vertex A at the origin. Let properties that you have learned about rectangles to the length of the bases be a units and the height be c walk through the proof. units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the Given: ABCD is a rectangle. proof step by step. Look over what you are given Prove: and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Prove: ABCD is a rectangle. Proof: Use the Distance Formula to find the lengths of the diagonals.

The diagonals, have the same length, so Proof:

they are congruent.

ANSWER: Given: ABCD is a rectangle.

Prove: AC = BD. So,

Proof: Because A and B are different points, Use the Distance Formula to find a ≠ 0. Then b = 0. The slope of is undefined and . and the slope of . Thus, . ∠DAB is

have the same length, so they are a right angle and ABCD is a rectangle. congruent. ANSWER: Given: ABCD and 43. If the diagonals of a parallelogram are congruent, Prove: ABCD is a rectangle. then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the Proof: proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the But AC = BD. So, proof.

Given: ABCD and Prove: ABCD is a rectangle.

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . DAB is Proof: ∠

a right angle and ABCD is a rectangle.. 44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. AC = BD. So, a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle. SOLUTION: a. Sample answer: ANSWER: Given: ABCD and Prove: ABCD is a rectangle.

Proof:

b. Use a protractor to measure each angle listed in the table. But AC = BD. So,

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: Because A and B are different points, a. Sample answer: a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

c. VERBAL Make a conjecture about the diagonals c. Sample answer: The diagonals of a parallelogram of a parallelogram with four congruent sides. with four congruent sides are perpendicular.

SOLUTION: 45. CHALLENGE In rectangle ABCD, a. Sample answer: Find the values of x and y.

SOLUTION: All four angles of a rectangle are right angles. So, b. Use a protractor to measure each angle listed in

the table.

The diagonals of a rectangle are congruent and c. Sample answer: Each of the angles listed in the bisect each other. So, is an isosceles triangle table are right angles. These angles were each formed by diagonals. The diagonals of a with parallelogram with four congruent sides are So, perpendicular.

ANSWER: a. Sample answer: The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

ANSWER: x = 6, y = −10

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and So, Tamika is correct. bisect each other. So, is an isosceles triangle ANSWER: with Tamika; Sample answer: When two congruent So, triangles are arranged to form a quadrilateral, two of

the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle. The sum of the angles of a triangle is 180. REASONING 47. In the diagram at the right, lines n, p, So, q,and r are parallel and lines l and m are parallel. By the Vertical Angle Theorem, How many rectangles are formed by the intersecting lines?

ANSWER: x = 6, y = −10

In order for the quadrilateral to be a rectangle, one of

the angles in the congruent triangles has to be a right Then the rectangles formed are ABXW, ACYW, angle. BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4; Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the SOLUTION: slope of DC is 0. Since a pair of sides of the Sample answer: quadrilateral is both parallel and congruent, by A rectangle has 4 right angles and each pair of Theorem 6.12, the quadrilateral is a parallelogram. opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other Since is horizontal and is vertical, the lines and perpendicular to lines parallel to the y-axis. are perpendicular and the measure of the angle they Selecting y = 0 and x = 0 will form 2 sides of the form is 90. By Theorem 6.6, if a parallelogram has rectangle. The lines y = 4 and x = 6 form the other 2 one right angle, it has four right angles. Therefore, by sides. definition, the parallelogram is a rectangle.

49. WRITING IN MATH Explain why all rectangles x = 0, x = 6, y = 0, y = 4; are parallelograms, but all parallelograms are not rectangles. SOLUTION: Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms.

Parallelograms with right angles are rectangles, so The length of is 6 − 0 or 6 units and the length of some parallelograms are rectangles, but others with is 6 – 0 or 6 units. The slope of AB is 0 and the non-right angles are not. slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by ANSWER: Theorem 6.12, the quadrilateral is a parallelogram. Sample answer: All rectangles are parallelograms Since is horizontal and is vertical, the lines because, by definition, both pairs of opposite sides are perpendicular and the measure of the angle they are parallel. Parallelograms with right angles are form is 90. By Theorem 6.6, if a parallelogram has rectangles, so some parallelograms are rectangles, one right angle, it has four right angles. Therefore, by but others with non-right angles are not. definition, the parallelogram is a rectangle. 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = ANSWER: 13, what values of x and y make parallelogram

Sample answer: x = 0, x = 6, y = 0, FGHJ a rectangle? y = 4;

A x = 3, y = 4 B x = 4, y = 3 C x = 7, y = 8 D x = 8, y = 7

SOLUTION: The length of is 6 − 0 or 6 units and the length of The opposite sides of a parallelogram are congruent. So, FJ = GH. is 6 – 0 or 6 units. The slope of AB is 0 and the –3x + 5y = 11 FJ = GH slope of DC is 0. Since a pair of sides of the

quadrilateral is both parallel and congruent, by If the diagonals of a parallelogram are congruent, Theorem 6.12, the quadrilateral is a parallelogram. then it is a rectangle. Since FGHJ is a parallelogram, Since is horizontal and is vertical, the lines the diagonals bisect each other. So, if it is a are perpendicular and the measure of the angle they rectangle, then will be an isosceles triangle form is 90. By Theorem 6.6, if a parallelogram has with That is, 3x + y = 13. one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. Add the two equations to eliminate the x-term and

49. WRITING IN MATH Explain why all rectangles solve for y.

are parallelograms, but all parallelograms are not –3x + 5y = 11

rectangles. 3x + y = 13 6y = 24 Add to eliminate x-term. SOLUTION: y = 4 Divide each side by 6. Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By Use the value of y to find the value of x. definition of rectangle, all rectangles have both pairs 3x + 4 = 13 Substitute. of opposite sides parallel. So rectangles are always 3x = 9 Subtract 4 from each side. parallelograms. x = 3 Divide each side by 3.

Parallelograms with right angles are rectangles, so So, x = 3, y = 4, the correct choice is A. some parallelograms are rectangles, but others with non-right angles are not. ANSWER: A ANSWER: Sample answer: All rectangles are parallelograms 51. ALGEBRA A rectangular playground is surrounded because, by definition, both pairs of opposite sides by an 80-foot fence. One side of the playground is 10 are parallel. Parallelograms with right angles are feet longer than the other. Which of the following rectangles, so some parallelograms are rectangles, equations could be used to find r, the shorter side of but others with non-right angles are not. the playground? F 10r + r = 80 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = G 4r + 10 = 80 13, what values of x and y make parallelogram H r(r + 10) = 80 FGHJ a rectangle? J 2(r + 10) + 2r = 80 SOLUTION: The formula to find the perimeter of a rectangle is given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 A x = 3, y = 4 (r + 10) = 80. B x = 4, y = 3 Therefore, the correct choice is J. C x = 7, y = 8 D x = 8, y = 7 ANSWER: J SOLUTION: The opposite sides of a parallelogram are congruent. 52. SHORT RESPONSE What is the measure of So, FJ = GH. ? –3x + 5y = 11 FJ = GH

51. ALGEBRA A rectangular playground is surrounded D by an 80-foot fence. One side of the playground is 10 E p + 2 feet longer than the other. Which of the following SOLUTION: equations could be used to find r, the shorter side of Analyze each answer choice to determine which the playground? expression is odd when p is odd. F 10r + r = 80 Choice A 2p G 4r + 10 = 80 This is a multiple of 2 so it is even. H r(r + 10) = 80 Choice B 2p + 2 J 2(r + 10) + 2r = 80 This can be rewritten as 2(p + 1) which is also a SOLUTION: multiple of 2 so it is even.

The formula to find the perimeter of a rectangle is Choice C given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 If p is odd , is not a whole number. (r + 10) = 80. Choice D 2p – 2 Therefore, the correct choice is J. This can be rewritten as 2(p - 1) which is also a ANSWER: multiple of 2 so it is even. Choice E p + 2 J Adding 2 to an odd number results in an odd 52. SHORT RESPONSE What is the measure of number. This is odd. ? The correct choice is E. ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with 54. So, SOLUTION: By the Exterior Angle Theorem, The opposite sides of a parallelogram are congruent.

ANSWER: 112 In a parallelogram, consecutive interior angles are 53. SAT/ACT If p is odd, which of the following must supplementary. also be odd? A 2p B 2p + 2

C D E p + 2 ANSWER: x = 2, y = 41 SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a 55. multiple of 2 so it is even. SOLUTION: Choice C The opposite sides of a parallelogram are congruent. If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. The opposite sides of a parallelogram are parallel to Choice E p + 2 each other. So, the angles with the measures (3y + Adding 2 to an odd number results in an odd 3)° and (4y – 19)° are alternate interior angles and

number. This is odd. hence they are equal.

The correct choice is E.

ANSWER: E ANSWER: ALGEBRA Find x and y so that the x = 8, y = 22 quadrilateral is a parallelogram.

2y = 14 In a parallelogram, consecutive interior angles are y = 7 supplementary. Use the value of y in the second equation and solve

for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: x = 8, y = 22

58. If , name two congruent angles. SOLUTION: 56. Since is an isosceles triangle. So,

SOLUTION: The diagonals of a parallelogram bisect each other. ANSWER: So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 59. If , name two congruent segments. Use the value of y in the second equation and solve SOLUTION: for x. 7 + 3 = 2x + 6 In a triangle with two congruent angles, the sides 4 = 2x opposite to the congruent angles are also congruent. 2 = x So, .

ANSWER: ANSWER: x = 2, y = 7

The diagonals of a parallelogram bisect each other, So, . so the intersection is the midpoint of either diagonal. ANSWER: Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and 1. QR SOLUTION: , find each measure. The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft ANSWER: 2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So, 3.

SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Therefore, ANSWER: ANSWER: 33.5

3. 4. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. bisect each other. So, is an isosceles triangle. Then, Then, By the Exterior Angle Theorem, By the Vertical Angle Theorem,

Therefore, Therefore,

ANSWER: ANSWER: 56.5 33.5 ALGEBRA Quadrilateral DEFG is a rectangle. 4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem, 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: Therefore, The diagonals of a rectangle are congruent to each other. So, FD = EG. ANSWER: 56.5

ALGEBRA Quadrilateral DEFG is a rectangle. Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 11

5. If FD = 3x – 7 and EG = x + 5, find EG. 6. If , SOLUTION: find . The diagonals of a rectangle are congruent to each SOLUTION: other. So, FD = EG. All four angles of a rectangle are right angles. So,

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 11 ANSWER: 6. If , 51 find . 7. PROOF If ABDE is a rectangle and , SOLUTION: prove that . All four angles of a rectangle are right angles. So,

SOLUTION: ANSWER: You need to walk through the proof step by step. 51 Look over what you are given and what you need to prove. You are given ABDE is a rectangle and PROOF 7. If ABDE is a rectangle and , . You need to prove . Use the prove that . properties that you have learned about rectangles to walk through the proof.

Given: ABDE is a rectangle; Prove:

walk through the proof. (Given)

2. ABDE is a parallelogram. (Def. of rectangle) Given: ABDE is a rectangle; 3. (Opp. sides of a .)

Prove: 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Statements(Reasons) Prove: 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) Statements(Reasons) 6. (SAS)

1. ABDE is a rectangle; . 7. (CPCTC) (Given) ANSWER: 2. ABDE is a parallelogram. (Def. of rectangle) Given: ABDE is a rectangle; 3. (Opp. sides of a .)

Prove: 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula Use the slope formula to find the slope of the sides of the quadrilateral. SOLUTION:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

Use the slope formula to find the slope of the sides of ANSWER: the quadrilateral. No; slope of , slope of , slope of

, and slope of . Slope of

slope of , and slope of slope of , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of

slope of , and slope of slope of , so WXYZ is not a parallelogram. Therefore, 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance WXYZ is not a rectangle. Formula SOLUTION:

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula First, Use the Distance formula to find the lengths of the sides of the quadrilateral. SOLUTION:

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

First, Use the Distance formula to find the lengths of So the diagonals are congruent. Thus, ABCD is a the sides of the quadrilateral. rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are FENCING X-braces are also used to provide congruent. Thus, ABCD is a rectangle. support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure.

10. BC FENCING X-braces are also used to provide SOLUTION: support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. measure. ANSWER: 2 ft

11. DB SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 10. BC Theorem, DB = BC + DC . BC = AD = 2 SOLUTION: DC = AB = 6 The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: 2 ft ANSWER: 6.3 ft 11. DB 12. SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: is a right triangle. By the Pythagorean The diagonals of a rectangle are congruent and 2 2 2 Theorem, DB = BC + DC . bisect each other. So, is an isosceles triangle. BC = AD = 2 Then, DC = AB = 6

By the Vertical Angle Theorem, ANSWER: 6.3 ft ANSWER: 12. 50 SOLUTION: 13. The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, is an isosceles The diagonals of a rectangle are congruent and triangle. bisect each other. So, is an isosceles Then, triangle.

Then, All four angles of a rectangle are right angles.

By the Vertical Angle Theorem,

ANSWER: ANSWER: 50 25 13. CCSS REGULARITY Quadrilateral WXYZ is a SOLUTION: rectangle. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, All four angles of a rectangle are right angles.

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: The opposite sides of a rectangle are parallel and ANSWER: congruent. So, ZY = WX. 25 2x + 3 = x + 4 x = 1 CCSS REGULARITY Quadrilateral WXYZ is a Use the value of x to find WX. rectangle. WX = 1 + 4 = 5.

ANSWER: 5

15. If PY = 3x – 5 and WP = 2x + 11, find ZP.

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: SOLUTION: The diagonals of a rectangle bisect each other. So, The opposite sides of a rectangle are parallel and PY = WP. congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 The diagonals of a rectangle are congruent and Use the value of x to find WX. bisect each other. So, is an isosceles triangle. WX = 1 + 4 = 5.

ANSWER: 5 ANSWER: 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. 43 SOLUTION: 16. If , find The diagonals of a rectangle bisect each other. So, . PY = WP. SOLUTION: All four angles of a rectangle are right angles. So, The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

ANSWER: 43 ANSWER: 16. If , find 39 . 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY.

Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38.

ANSWER: ANSWER: 38 39 18. If , find 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. . SOLUTION: The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, ZP = PY. All four angles of a rectangle are right angles. So,

Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38.

ANSWER: ANSWER: 38 48

18. If , find 19. If , find

. . SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, All four angles of a rectangle are right angles. So,

is a right triangle. Then,

ANSWER: Therefore, Since is a 48 transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, 19. If , find = = 46. . ANSWER: SOLUTION: 46 All four angles of a rectangle are right angles. So, is a right triangle. Then, PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. Prove:

Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, SOLUTION: = = 46. You need to walk through the proof step by step. Look over what you are given and what you need to ANSWER: prove. Here, you are given ABCD is a rectangle. 46 You need to prove . Use the properties that you have learned about rectangles to PROOF Write a two-column proof. walk through the proof. 20. Given: ABCD is a rectangle. Prove: Given: ABCD is a rectangle. Prove: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) SOLUTION: 3. (Opp. sides of a .) You need to walk through the proof step by step. 4. (Refl. Prop.) Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. 5. (Diagonals of a rectangle are .) You need to prove . Use the 6. (SSS) properties that you have learned about rectangles to ANSWER: walk through the proof. Proof: Statements (Reasons) Given: ABCD is a rectangle. 1. ABCD is a rectangle. (Given) Prove: 2. ABCD is a parallelogram. (Def. of rectangle) Proof: Statements (Reasons) 3. (Opp. sides of a .) 1. ABCD is a rectangle. (Given) 4. (Refl. Prop.) 2. ABCD is a parallelogram. (Def. of rectangle) 5. (Diagonals of a rectangle are .) 3. (Opp. sides of a .) 6. (SSS) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 21. Given: QTVW is a rectangle. 6. (SSS)

ANSWER: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) Prove: 3. (Opp. sides of a .) SOLUTION: 4. (Refl. Prop.) You need to walk through the proof step by step.

5. (Diagonals of a rectangle are .) Look over what you are given and what you need to 6. (SSS) prove. Here, you are given QTVW is a rectangle and . You need to prove . Use 21. Given: QTVW is a rectangle. the properties that you have learned about rectangles to walk through the proof.

Given: QTVW is a rectangle.

Prove:

SOLUTION: You need to walk through the proof step by step. Prove: Look over what you are given and what you need to Proof: prove. Here, you are given QTVW is a rectangle and Statements (Reasons) . You need to prove . Use 1. QTVW is a rectangle; .(Given) the properties that you have learned about rectangles 2. QTVW is a parallelogram. (Def. of rectangle) to walk through the proof. 3. (Opp sides of a .)

Given: QTVW is a rectangle. 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) Prove: 11. QS = RT (Subst.) Proof: Statements (Reasons) 12. (Def. of segs.)

1. QTVW is a rectangle; .(Given) 13. (SAS) 2. QTVW is a parallelogram. (Def. of rectangle) ANSWER:

3. (Opp sides of a .) Proof: 4. are right angles.(Def. of rectangle) Statements (Reasons) 5. (All rt .) 1. QTVW is a rectangle; .(Given) 6. QR = ST (Def. of segs.) 2. QTVW is a parallelogram. (Def. of rectangle) 7. (Refl. Prop.) 3. (Opp sides of a .) 8. RS = RS (Def. of segs.) 4. are right angles.(Def. of rectangle) 9. QR + RS = RS + ST (Add. prop.) 5. (All rt .) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 6. QR = ST (Def. of segs.) 11. QS = RT (Subst.) 7. (Refl. Prop.) 12. (Def. of segs.) 8. RS = RS (Def. of segs.)

13. (SAS) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) ANSWER: 11. QS = RT (Subst.) Proof: Statements (Reasons) 12. (Def. of segs.)

1. QTVW is a rectangle; .(Given) 13. (SAS)

2. QTVW is a parallelogram. (Def. of rectangle) COORDINATE GEOMETRY Graph each 3. (Opp sides of a .) quadrilateral with the given vertices. Determine 4. are right angles.(Def. of rectangle) whether the figure is a rectangle. Justify your 5. (All rt .) answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) SOLUTION: 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. Use the slope formula to find the slope of the sides of The products of the slopes of the adjacent sides are the quadrilateral. –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle.

ANSWER:

Yes; slope of slope of , slope

of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are The slopes of each pair of opposite sides are equal. perpendicular and form right angles. Thus, WXYZ is a So, the two pairs of opposite sides are parallel. rectangle. Therefore, the quadrilateral WXYZ is a parallelogram. The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle.

ANSWER:

Yes; slope of slope of , slope

of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle. 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula First, Use the Distance formula to find the lengths of SOLUTION: the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

First, Use the Distance formula to find the lengths of So the diagonals are not congruent. Thus, JKLM is the sides of the quadrilateral. not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so so JKLM is a the diagonals are not congruent. Thus, JKLM is not a

parallelogram. rectangle. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a 6-4 Rectangles parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle. 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula First, use the Distance Formula to find the lengths of the sides of the quadrilateral. SOLUTION:

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

First, use the Distance Formula to find the lengths of So the diagonals are congruent. Thus, QRST is a the sides of the quadrilateral. rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are so QRST is a congruent. QRST is a rectangle. parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula congruent. QRST is a rectangle. SOLUTION: eSolutions Manual - Powered by Cognero Page 8

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula

SOLUTION: First, use the slope formula to find the lengths of the sides of the quadrilateral.

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a First, use the slope formula to find the lengths of the parallelogram. sides of the quadrilateral. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle.

ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are The slopes of each pair of opposite sides are equal. not perpendicular. Thus, GHJK is not a rectangle. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle.

ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle. Quadrilateral ABCD is a rectangle. Find each measure if .

26. SOLUTION: All four angles of a rectangle are right angles. So,

Quadrilateral ABCD is a rectangle. Find each measure if . ANSWER: 50

27.

SOLUTION: 26. The measures of angles 2 and 3 are equal as they are alternate interior angles. SOLUTION: Since the diagonals of a rectangle are congruent and All four angles of a rectangle are right angles. So, bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: ANSWER: 40 50 28.

27. SOLUTION: The measures of angles 2 and 3 are equal as they SOLUTION: are alternate interior angles. The measures of angles 2 and 3 are equal as they Therefore, m∠3 = m∠2 = 40. are alternate interior angles. Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 3, 7 40 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. 29. ANSWER: SOLUTION: 40 All four angles of a rectangle are right angles. So,

28. SOLUTION: The measures of angles 2 and 3 are equal as they The measures of angles 1 and 4 are equal as they

are alternate interior angles. are alternate interior angles. Therefore, m 4 = m 1 = 50. Therefore, m∠3 = m∠2 = 40. ∠ ∠ Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 4, 5 40 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. 29. Therefore, m∠5 = 180 – (50 + 50) = 80. SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 80

30. SOLUTION: The measures of angles 1 and 4 are equal as they All four angles of a rectangle are right angles. So, are alternate interior angles.

Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m 6 = m 4 = 50. The measures of angles 1 and 4 are equal as they ∠ ∠ The sum of the three angles of a triangle is 180. are alternate interior angles. Therefore, m 4 = m 1 = 50. Therefore, m∠5 = 180 – (50 + 50) = 80. ∠ ∠ Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 4, 5 80 and 6 is an isosceles triangle. Therefore, m∠6 = m∠4 = 50. 30. ANSWER: SOLUTION: 50 All four angles of a rectangle are right angles. So, 31. SOLUTION: The measures of angles 2 and 3 are equal as they The measures of angles 1 and 4 are equal as they are alternate interior angles. are alternate interior angles. Since the diagonals of a rectangle are congruent and Therefore, m∠4 = m∠1 = 50. bisect each other, the triangle with the angles 3, 7 Since the diagonals of a rectangle are congruent and and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. bisect each other, the triangle with the angles 4, 5 The sum of the three angles of a triangle is 180. and 6 is an isosceles triangle. Therefore, m∠6 = Therefore, m∠8 = 180 – (40 + 40) = 100. m∠4 = 50. ANSWER: ANSWER: 100 50 32. CCSS MODELING Jody is building a new 31. bookshelf using wood and metal supports like the one SOLUTION: shown. To what length should she cut the metal supports in order for the bookshelf to be square, The measures of angles 2 and 3 are equal as they which means that the angles formed by the shelves

are alternate interior angles. and the vertical supports are all right angles? Explain Since the diagonals of a rectangle are congruent and your reasoning. bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. Therefore, m∠8 = 180 – (40 + 40) = 100. ANSWER: 100

32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one SOLUTION: shown. To what length should she cut the metal supports in order for the bookshelf to be square, In order for the bookshelf to be square, the lengths of which means that the angles formed by the shelves all of the metal supports should be equal. Since we and the vertical supports are all right angles? Explain know the length of the bookshelves and the distance your reasoning. between them, use the Pythagorean Theorem to find the lengths of the metal supports. Let x be the length of each support. Express 3 feet as 36 inches.

Therefore, the metal supports should each be 39 SOLUTION: inches long or 3 feet 3 inches long. In order for the bookshelf to be square, the lengths of ANSWER: all of the metal supports should be equal. Since we know the length of the bookshelves and the distance 3 ft 3 in.; Sample answer: In order for the bookshelf between them, use the Pythagorean Theorem to find to be square, the lengths of all of the metal supports the lengths of the metal supports. should be equal. Since I knew the length of the Let x be the length of each support. Express 3 feet bookshelves and the distance between them, I used as 36 inches. the Pythagorean Theorem to find the lengths of the metal supports. PROOF Write a two-column proof. 33. Theorem 6.13 SOLUTION: Therefore, the metal supports should each be 39 You need to walk through the proof step by step. inches long or 3 feet 3 inches long. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with ANSWER: diagonals . You need to prove 3 ft 3 in.; Sample answer: In order for the bookshelf . Use the properties that you have learned to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the about rectangles to walk through the proof. bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the Given: WXYZ is a rectangle with diagonals metal supports. .

PROOF Write a two-column proof. Prove: 33. Theorem 6.13 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with Proof: diagonals . You need to prove 1. WXYZ is a rectangle with diagonals . . Use the properties that you have learned (Given) about rectangles to walk through the proof. 2. (Opp. sides of a .) 3. (Refl. Prop.) Given: WXYZ is a rectangle with diagonals 4. are right angles. (Def. of

. rectangle) Prove: 5. (All right .) 6. (SAS) 7. (CPCTC)

ANSWER:

Proof: Given: WXYZ is a rectangle with diagonals

1. WXYZ is a rectangle with diagonals . . (Given) Prove: 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) 5. (All right .) Proof: 6. (SAS) 1. WXYZ is a rectangle with diagonals . 7. (CPCTC) (Given) 2. (Opp. sides of a .) ANSWER: 3. (Refl. Prop.) Given: WXYZ is a rectangle with diagonals 4. are right angles. (Def. of . rectangle) Prove: 5. (All right .) 6. (SAS) 7. (CPCTC)

34. Theorem 6.14 Proof: SOLUTION: 1. WXYZ is a rectangle with diagonals . You need to walk through the proof step by step.

(Given) Look over what you are given and what you need to 2. (Opp. sides of a .) prove. Here, you are given 3. (Refl. Prop.) . You need to 4. are right angles. (Def. of prove that WXYZ is a rectangle.. Use the properties rectangle) that you have learned about rectangles to walk 5. (All right .) through the proof.

6. (SAS) Given: 7. (CPCTC) Prove: WXYZ is a rectangle.

34. Theorem 6.14 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given Proof:

. You need to 1. (Given) prove that WXYZ is a rectangle.. Use the properties 2. (SSS) that you have learned about rectangles to walk 3. (CPCTC) through the proof. 4. (Def. of )

5. WXYZ is a parallelogram. (If both pairs of opp.

Given: sides are , then quad. is .) Prove: WXYZ is a rectangle. 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .)

Proof: 9. are right angles. (If a has 1 1. (Given) rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) 2. (SSS) 3. (CPCTC) ANSWER: 4. (Def. of ) Given: 5. WXYZ is a parallelogram. (If both pairs of opp. Prove: WXYZ is a rectangle. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) Proof:

9. are right angles. (If a has 1 1. (Given) rt. , it has 4 rt. .) 2. (SSS) 10. WXYZ is a rectangle. (Def. of rectangle) 3. (CPCTC) 4. (Def. of ) ANSWER: 5. WXYZ is a parallelogram. (If both pairs of opp.

Given: sides are , then quad. is .) Prove: WXYZ is a rectangle. 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .)

Proof: 9. are right angles. (If a has 1 1. (Given) rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) 2. (SSS) 3. (CPCTC) PROOF Write a paragraph proof of each 4. (Def. of ) statement. 5. WXYZ is a parallelogram. (If both pairs of opp. 35. If a parallelogram has one right angle, then it is a sides are , then quad. is .) rectangle. 6. are supplementary. (Cons. SOLUTION: are suppl.) You need to walk through the proof step by step. 7. (Def of suppl.) Look over what you are given and what you need to 8. are right angles. (If prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. 2 and suppl., each is a rt. .) Let ABCD be a parallelogram with one right angle. 9. are right angles. (If a has 1 Then use the properties of parallelograms to walk rt. , it has 4 rt. .) through the proof. 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION:

You need to walk through the proof step by step. ABCD is a parallelogram, and is a right angle. Look over what you are given and what you need to Since ABCD is a parallelogram and has one right prove. Here, you are given a parallelogram with one angle, then it has four right angles. So by the

right angle. You need to prove that it is a rectangle. definition of a rectangle, ABCD is a rectangle. Let ABCD be a parallelogram with one right angle. ANSWER: Then use the properties of parallelograms to walk through the proof.

ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the

ABCD is a parallelogram, and is a right angle. definition of a rectangle, ABCD is a rectangle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. 36. If a quadrilateral has four right angles, then it is a rectangle. ANSWER: SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and ABCD is a parallelogram, and is a right angle. rectangles to walk through the proof. Since ABCD is a parallelogram and has one right

angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle.

36. If a quadrilateral has four right angles, then it is a rectangle. SOLUTION: ABCD is a quadrilateral with four right angles. You need to walk through the proof step by step. ABCD is a parallelogram because both pairs of Look over what you are given and what you need to opposite angles are congruent. By definition of a prove. Here, you are given a quadrilateral with four rectangle, ABCD is a rectangle. right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. ANSWER: Then use the properties of parallelograms and rectangles to walk through the proof.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a

ABCD is a quadrilateral with four right angles. rectangle, ABCD is a rectangle. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a 37. CONSTRUCTION Construct a rectangle using the rectangle, ABCD is a rectangle. construction for congruent segments and the ANSWER: construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. SOLUTION: Sample answer: Step 1: Graph a line and place points P and Q on the line. ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of Step 2: Place the compass on point P. Using the opposite angles are congruent. By definition of a same compass setting, draw an arc to the left and rectangle, ABCD is a rectangle. right of P that intersects . 37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the Step 3: Open the compass to a setting greater than construction for a line perpendicular to another line the distance from P to either point of intersection. through a point on the line. Justify each step of the Place the compass on each point of intersection and

construction. draw arcs that intersect above . SOLUTION: Sample answer: Step 4: Draw line a through point P and the Step 1: Graph a line and place points P and Q on the intersection of the arcs drawn in step 3. line. Step 5: Repeat steps 2 and 3 using point Q. Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5. right of P that intersects .

Step 7: Using any compass setting, put the compass Step 3: Open the compass to a setting greater than at P and draw an arc above it that intersects line a. the distance from P to either point of intersection. Label the point of intersection R. Place the compass on each point of intersection and

draw arcs that intersect above . Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that Step 4: Draw line a through point P and the intersects line b. Label the point of intersection S. intersection of the arcs drawn in step 3.

Step 9: Draw . Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S. Since and , the measure of angles P and Q is 90 degrees. Lines that are Step 9: Draw . perpendicular to the same line are parallel, so . The same compass setting was used to locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

ANSWER:

Since and , the measure of Sample answer: Since and , the angles P and Q is 90 degrees. Lines that are measure of angles P and Q is 90 degrees. Lines that perpendicular to the same line are parallel, so are perpendicular to the same line are parallel, so . The same compass setting was used to . The same compass setting was used to locate points R and S, so . If one pair of locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Thus, PRSQ is a rectangle.

ANSWER:

Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so . The same compass setting was used to locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. 38. SPORTS The end zone of a football field is 160 feet Thus, PRSQ is a rectangle. wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then the end zone is a rectangle. 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for ANSWER: painting the field. He has finished the end zone. Sample answer: He should measure the diagonals of Explain how Kyle can confirm that the end zone is the end zone and both pairs of opposite sides. If the the regulation size and be sure that it is also a diagonals and both pairs of opposite sides are rectangle using only a tape measure. congruent, then the end zone is a rectangle.

SOLUTION: ALGEBRA Quadrilateral WXYZ is a rectangle. Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then 39. If XW = 3, WZ = 4, and XZ = b, find YW.

the end zone is a rectangle. SOLUTION: ANSWER: The diagonals of a rectangle are congruent to each Sample answer: He should measure the diagonals of other. So, YW = XZ = b. All four angles of a the end zone and both pairs of opposite sides. If the rectangle are right angles. So, is a right diagonals and both pairs of opposite sides are triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 congruent, then the end zone is a rectangle. WZ . ALGEBRA Quadrilateral WXYZ is a rectangle.

Therefore, YW = 5.

ANSWER:

39. If XW = 3, WZ = 4, and XZ = b, find YW. 5 SOLUTION: 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a SOLUTION: rectangle are right angles. So, is a right The diagonals of a rectangle are congruent to each triangle. By the Pythagorean Theorem, XZ2 = XW 2 + other. So, WY = XZ = 2c. All four angles of a 2 rectangle are right angles. So, is a right WZ . 2 2 triangle. By the Pythagorean Theorem, XZ = XY + 2 ZY .

Therefore, YW = 5.

ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. Therefore, WY = 2(5) = 10. SOLUTION: ANSWER: The diagonals of a rectangle are congruent to each 10 other. So, WY = XZ = 2c. All four angles of a rectangle are right angles. So, is a right 41. SIGNS The sign below is in the foyer of Nyoko’s 2 2 school. Based on the dimensions given, can Nyoko triangle. By the Pythagorean Theorem, XZ = XY + be sure that the sign is a rectangle? Explain your 2 ZY . reasoning.

Therefore, WY = 2(5) = 10.

ANSWER: 10 SOLUTION:

41. SIGNS The sign below is in the foyer of Nyoko’s Both pairs of opposite sides are congruent, so the school. Based on the dimensions given, can Nyoko sign is a parallelogram, but no measure is given that be sure that the sign is a rectangle? Explain your can be used to prove that it is a rectangle. If the reasoning. diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. SOLUTION: PROOF Write a coordinate proof of each statement. Both pairs of opposite sides are congruent, so the 42. The diagonals of a rectangle are congruent. sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the SOLUTION: diagonals could be proven to be congruent, or if the Begin by positioning quadrilateral ABCD on a angles could be proven to be right angles, then we coordinate plane. Place vertex D at the origin. Let could prove that the sign was a rectangle. the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, ANSWER: b), and C(a, 0). You need to walk through the proof No; sample answer: Both pairs of opposite sides are step by step. Look over what you are given and what congruent, so the sign is a parallelogram, but no you need to prove. Here, you are given ABCD is a measure is given that can be used to prove that it is a rectangle and you need to prove . Use the rectangle. properties that you have learned about rectangles to walk through the proof. PROOF Write a coordinate proof of each statement. Given: ABCD is a rectangle. 42. The diagonals of a rectangle are congruent. Prove: SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a Proof: rectangle and you need to prove . Use the Use the Distance Formula to find the lengths of the properties that you have learned about rectangles to diagonals. walk through the proof.

Given: ABCD is a rectangle. Prove: The diagonals, have the same length, so they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove:

Proof: Use the Distance Formula to find the lengths of the diagonals.

The diagonals, have the same length, so Proof: Use the Distance Formula to find they are congruent. and . ANSWER: have the same length, so they are Given: ABCD is a rectangle. congruent. Prove: 43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + Proof: a, c), and D(b, c). You need to walk through the Use the Distance Formula to find proof step by step. Look over what you are given and . and what you need to prove. Here, you are given have the same length, so they are ABCD and and you need to prove that congruent. ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof. 43. If the diagonals of a parallelogram are congruent,

then it is a rectangle. Given: ABCD and SOLUTION: Prove: ABCD is a rectangle. Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that Proof: ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and AC = BD. So,

Prove: ABCD is a rectangle.

Proof: Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.

AC = BD. So, ANSWER: Given: ABCD and Prove: ABCD is a rectangle.

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined Proof: and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle. ANSWER: But AC = BD. So,

Given: ABCD and Prove: ABCD is a rectangle.

Because A and B are different points, Proof: a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

But AC = BD. So, 44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the Because A and B are different points, appropriate angles and complete the table below. a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.. c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. 44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special SOLUTION: parallelograms. a. Sample answer: a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: b. Use a protractor to measure each angle listed in a. Sample answer: the table.

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: a. Sample answer:

b. Use a protractor to measure each angle listed in the table.

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: b. a. Sample answer:

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular. SOLUTION: 45. CHALLENGE In rectangle ABCD, All four angles of a rectangle are right angles. So, Find the values of x and y.

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So,

SOLUTION: The sum of the angles of a triangle is 180. All four angles of a rectangle are right angles. So, So, By the Vertical Angle Theorem,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So, ANSWER:

x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a The sum of the angles of a triangle is 180. rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is So, either of them correct? Explain your reasoning. By the Vertical Angle Theorem, SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a In order for the quadrilateral to be a rectangle, one of rectangle. Tamika says that only two congruent right the angles in the congruent triangles has to be a right triangles can be arranged to make a rectangle. Is angle. either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

So, Tamika is correct.

ANSWER: Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of In order for the quadrilateral to be a rectangle, one of the angles are formed by a single vertex of a triangle. the angles in the congruent triangles has to be a right In order for the quadrilateral to be a rectangle, one of angle. the angles in the congruent triangles has to be a right angle.

47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines?

So, Tamika is correct.

ANSWER: Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle. SOLUTION: Let the lines n, p, q, and r intersect the line l at the 47. REASONING In the diagram at the right, lines n, p, points A, B, C, and D respectively. Similarly, let the q,and r are parallel and lines l and m are parallel. lines intersect the line m at the points W, X, Y, and Z. How many rectangles are formed by the intersecting lines?

SOLUTION: Let the lines n, p, q, and r intersect the line l at the Then the rectangles formed are ABXW, ACYW, points A, B, C, and D respectively. Similarly, let the BCYX, BDZX, CDZY, and ADZW. Therefore, 6 lines intersect the line m at the points W, X, Y, and Z. rectangles are formed by the intersecting lines.

ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of Then the rectangles formed are ABXW, ACYW, opposite sides is parallel and congruent. Lines that BCYX, BDZX, CDZY, and ADZW. Therefore, 6 run parallel to the x-axis will be parallel to each other rectangles are formed by the intersecting lines. and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the ANSWER: rectangle. The lines y = 4 and x = 6 form the other 2 6 sides.

48. OPEN ENDED Write the equations of four lines x = 0, x = 6, y = 0, y = 4; having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the The length of is 6 − 0 or 6 units and the length of rectangle. The lines y = 4 and x = 6 form the other 2 sides. is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by x = 0, x = 6, y = 0, y = 4; Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4; The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. The length of is 6 − 0 or 6 units and the length of ANSWER: is 6 – 0 or 6 units. The slope of AB is 0 and the Sample answer: x = 0, x = 6, y = 0, slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by y = 4; Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not rectangles. The length of is 6 − 0 or 6 units and the length of SOLUTION: is 6 – 0 or 6 units. The slope of AB is 0 and the Sample answer: A parallelogram is a quadrilateral in slope of DC is 0. Since a pair of sides of the which each pair of opposite sides are parallel. By quadrilateral is both parallel and congruent, by definition of rectangle, all rectangles have both pairs Theorem 6.12, the quadrilateral is a parallelogram. of opposite sides parallel. So rectangles are always Since is horizontal and is vertical, the lines parallelograms. are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has Parallelograms with right angles are rectangles, so one right angle, it has four right angles. Therefore, by some parallelograms are rectangles, but others with definition, the parallelogram is a rectangle. non-right angles are not.

49. WRITING IN MATH Explain why all rectangles ANSWER: are parallelograms, but all parallelograms are not Sample answer: All rectangles are parallelograms rectangles. because, by definition, both pairs of opposite sides are parallel. Parallelograms with right angles are SOLUTION: rectangles, so some parallelograms are rectangles, Sample answer: A parallelogram is a quadrilateral in but others with non-right angles are not. which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = of opposite sides parallel. So rectangles are always 13, what values of x and y make parallelogram parallelograms. FGHJ a rectangle?

Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

ANSWER: A x = 3, y = 4 Sample answer: All rectangles are parallelograms B x = 4, y = 3 because, by definition, both pairs of opposite sides C x = 7, y = 8 are parallel. Parallelograms with right angles are D x = 8, y = 7 rectangles, so some parallelograms are rectangles, but others with non-right angles are not. SOLUTION: The opposite sides of a parallelogram are congruent. 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = So, FJ = GH. 13, what values of x and y make parallelogram –3x + 5y = 11 FJ = GH FGHJ a rectangle?

If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a rectangle, then will be an isosceles triangle

A x = 3, y = 4 with That is, 3x + y = 13. B x = 4, y = 3 Add the two equations to eliminate the x-term and C x = 7, y = 8 solve for y. D x = 8, y = 7 –3x + 5y = 11 SOLUTION: 3x + y = 13 The opposite sides of a parallelogram are congruent. 6y = 24 Add to eliminate x-term. So, FJ = GH. y = 4 Divide each side by 6. –3x + 5y = 11 FJ = GH Use the value of y to find the value of x. If the diagonals of a parallelogram are congruent, 3x + 4 = 13 Substitute. then it is a rectangle. Since FGHJ is a parallelogram, 3x = 9 Subtract 4 from each side. the diagonals bisect each other. So, if it is a x = 3 Divide each side by 3. rectangle, then will be an isosceles triangle with That is, 3x + y = 13. So, x = 3, y = 4, the correct choice is A. ANSWER: Add the two equations to eliminate the x-term and A solve for y.

–3x + 5y = 11 51. ALGEBRA A rectangular playground is surrounded

3x + y = 13 by an 80-foot fence. One side of the playground is 10

6y = 24 Add to eliminate x-term. feet longer than the other. Which of the following y = 4 Divide each side by 6. equations could be used to find r, the shorter side of the playground?

Use the value of y to find the value of x. F 10r + r = 80 3x + 4 = 13 Substitute. G 4r + 10 = 80 3x = 9 Subtract 4 from each side. H r(r + 10) = 80 x = 3 Divide each side by 3. J 2(r + 10) + 2r = 80 So, x = 3, y = 4, the correct choice is A. SOLUTION: ANSWER: The formula to find the perimeter of a rectangle is given by P = 2(l + w). The perimeter is given to be 8 A ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. 51. ALGEBRA A rectangular playground is surrounded Therefore, the correct choice is J. by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following ANSWER: equations could be used to find r, the shorter side of J the playground? F 10r + r = 80 52. SHORT RESPONSE What is the measure of G 4r + 10 = 80 ? H r(r + 10) = 80 J 2(r + 10) + 2r = 80 SOLUTION: The formula to find the perimeter of a rectangle is given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. SOLUTION: Therefore, the correct choice is J. The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle with J So, 52. SHORT RESPONSE What is the measure of By the Exterior Angle Theorem, ? ANSWER: 112

53. SAT/ACT If p is odd, which of the following must also be odd? A 2p SOLUTION: B 2p + 2 The diagonals of a rectangle are congruent and C bisect each other. So, is an isosceles triangle with D So, E p + 2 By the Exterior Angle Theorem, SOLUTION: Analyze each answer choice to determine which ANSWER: expression is odd when p is odd. 112 Choice A 2p This is a multiple of 2 so it is even. 53. SAT/ACT If p is odd, which of the following must Choice B 2p + 2 also be odd? This can be rewritten as 2(p + 1) which is also a A 2p multiple of 2 so it is even. B 2p + 2 Choice C C If p is odd , is not a whole number. D Choice D 2p – 2 E p + 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. SOLUTION: Choice E p + 2 Analyze each answer choice to determine which Adding 2 to an odd number results in an odd expression is odd when p is odd. Choice A 2p number. This is odd. This is a multiple of 2 so it is even. The correct choice is E. Choice B 2p + 2 ANSWER: This can be rewritten as 2(p + 1) which is also a E multiple of 2 so it is even. ALGEBRA Find x and y so that the Choice C quadrilateral is a parallelogram. If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 54. Adding 2 to an odd number results in an odd SOLUTION: number. This is odd. The opposite sides of a parallelogram are congruent. The correct choice is E.

ANSWER: E In a parallelogram, consecutive interior angles are ALGEBRA Find x and y so that the supplementary. quadrilateral is a parallelogram.

54. ANSWER: SOLUTION: x = 2, y = 41 The opposite sides of a parallelogram are congruent.

In a parallelogram, consecutive interior angles are supplementary. 55. SOLUTION: The opposite sides of a parallelogram are congruent.

ANSWER: x = 2, y = 41 The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

55. ANSWER: SOLUTION: x = 8, y = 22 The opposite sides of a parallelogram are congruent.

The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + 56. 3)° and (4y – 19)° are alternate interior angles and hence they are equal. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 ANSWER: Use the value of y in the second equation and solve x = 8, y = 22 for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: 56. x = 2, y = 7 SOLUTION: 57. COORDINATE GEOMETRY Find the The diagonals of a parallelogram bisect each other. coordinates of the intersection of the diagonals of So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. parallelogram ABCD with vertices A(1, 3), B(6, 2), C Solve the first equation to find the value of y. (4, –2), and D(–1, –1). 2y = 14 y = 7 SOLUTION: Use the value of y in the second equation and solve The diagonals of a parallelogram bisect each other, for x. so the intersection is the midpoint of either diagonal. 7 + 3 = 2x + 6 Find the midpoint of or . 4 = 2x 2 = x

ANSWER: Therefore, the coordinates of the intersection of the x = 2, y = 7 diagonals are (2.5, 0.5).

57. COORDINATE GEOMETRY Find the ANSWER: coordinates of the intersection of the diagonals of (2.5, 0.5) parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). Refer to the figure. SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the

diagonals are (2.5, 0.5). 58. If , name two congruent angles.

ANSWER: SOLUTION: (2.5, 0.5) Since is an isosceles triangle. So,

Refer to the figure. ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent.

58. If , name two congruent angles. So, . SOLUTION: ANSWER: Since is an isosceles triangle. So,

ANSWER: 60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. 59. If , name two congruent segments. So, . SOLUTION: In a triangle with two congruent angles, the sides ANSWER: opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER: FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time.

If feet, PS = 7 feet, and 3. , find each measure. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Therefore,

ANSWER: 33.5 1. QR 4. SOLUTION: SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. ANSWER: Then, 7 ft By the Vertical Angle Theorem,

2. SQ Therefore, SOLUTION: The diagonals of a rectangle bisect each other. So, ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

5. If FD = 3x – 7 and EG = x + 5, find EG. ANSWER: SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

3. SOLUTION: The diagonals of a rectangle are congruent and Use the value of x to find EG. bisect each other. So, is an isosceles triangle. EG = 6 + 5 = 11 Then, ANSWER: By the Exterior Angle Theorem, 11

Therefore, 6. If , find . ANSWER: SOLUTION: 33.5 All four angles of a rectangle are right angles. So, 4.

SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

Then, By the Vertical Angle Theorem,

ANSWER:

Therefore, 51

ANSWER: 7. PROOF If ABDE is a rectangle and , 56.5 prove that . ALGEBRA Quadrilateral DEFG is a rectangle.

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent to each You need to walk through the proof step by step. other. So, FD = EG. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and . You need to prove . Use the properties that you have learned about rectangles to walk through the proof. Use the value of x to find EG. EG = 6 + 5 = 11 Given: ABDE is a rectangle; ANSWER: Prove: 11

6. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Statements(Reasons)

1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) ANSWER: 5. (All rt .) 51 6. (SAS) 7. (CPCTC) 7. PROOF If ABDE is a rectangle and , ANSWER: prove that . Given: ABDE is a rectangle; Prove:

SOLUTION: You need to walk through the proof step by step. Statements(Reasons) Look over what you are given and what you need to 1. ABDE is a rectangle; . prove. You are given ABDE is a rectangle and (Given) . You need to prove . Use the 2. ABDE is a parallelogram. (Def. of properties that you have learned about rectangles to rectangle) walk through the proof. 3. (Opp. sides of a .)

4. are right angles.(Def. of rectangle)

Given: ABDE is a rectangle; 5. (All rt .)

Prove: 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula Statements(Reasons) SOLUTION: 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER:

Given: ABDE is a rectangle; Use the slope formula to find the slope of the sides of

Prove: the quadrilateral.

Statements(Reasons) 1. ABDE is a rectangle; . (Given) There are no parallel sides, so WXYZ is not a 2. ABDE is a parallelogram. (Def. of parallelogram. Therefore, WXYZ is not a rectangle. rectangle) ANSWER: 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) No; slope of , slope of , slope of 5. (All rt .) 6. (SAS) , and slope of . Slope of

7. (CPCTC) slope of , and slope of slope of COORDINATE GEOMETRY Graph each , so WXYZ is not a parallelogram. Therefore, quadrilateral with the given vertices. Determine WXYZ is not a rectangle. whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION: Use the slope formula to find the slope of the sides of the quadrilateral.

There are no parallel sides, so WXYZ is not a First, Use the Distance formula to find the lengths of parallelogram. Therefore, WXYZ is not a rectangle. the sides of the quadrilateral.

ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of so ABCD is a parallelogram. slope of , and slope of slope of A parallelogram is a rectangle if the diagonals are , so WXYZ is not a parallelogram. Therefore, congruent. Use the Distance formula to find the WXYZ is not a rectangle. lengths of the diagonals.

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so ABCD is a 10. BC parallelogram. A parallelogram is a rectangle if the diagonals are SOLUTION: congruent. Use the Distance formula to find the The opposite sides of a rectangle are parallel and lengths of the diagonals. congruent. Therefore, BC = AD = 2 ft.

ANSWER: So the diagonals are congruent. Thus, ABCD is a 2 ft rectangle. 11. DB ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a SOLUTION: All four angles of a rectangle are right angles. So, parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

ANSWER: 6.3 ft

12.

SOLUTION: FENCING X-braces are also used to provide The diagonals of a rectangle are congruent and support in rectangular fencing. If AB = 6 feet, bisect each other. So, is an isosceles AD = 2 feet, and , find each triangle. measure. Then,

By the Vertical Angle Theorem,

ANSWER: 50

10. BC 13. SOLUTION: SOLUTION: The opposite sides of a rectangle are parallel and The diagonals of a rectangle are congruent and

congruent. Therefore, BC = AD = 2 ft. bisect each other. So, is an isosceles ANSWER: triangle.

2 ft Then, All four angles of a rectangle are right angles.

11. DB SOLUTION: All four angles of a rectangle are right angles. So, ANSWER: is a right triangle. By the Pythagorean 2 2 2 25 Theorem, DB = BC + DC . BC = AD = 2 CCSS REGULARITY Quadrilateral WXYZ is a DC = AB = 6 rectangle.

ANSWER: 6.3 ft 14. If ZY = 2x + 3 and WX = x + 4, find WX.

12. SOLUTION: SOLUTION: The opposite sides of a rectangle are parallel and The diagonals of a rectangle are congruent and congruent. So, ZY = WX. bisect each other. So, is an isosceles 2x + 3 = x + 4 triangle. x = 1 Then, Use the value of x to find WX. WX = 1 + 4 = 5. ANSWER: 5 By the Vertical Angle Theorem,

15. If PY = 3x – 5 and WP = 2x + 11, find ZP. ANSWER: SOLUTION: 50 The diagonals of a rectangle bisect each other. So,

13. PY = WP.

SOLUTION: The diagonals of a rectangle are congruent and The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles bisect each other. So, is an isosceles triangle. triangle. Then, All four angles of a rectangle are right angles.

ANSWER: 43

ANSWER: 16. If , find 25 . CCSS REGULARITY Quadrilateral WXYZ is a SOLUTION: rectangle. All four angles of a rectangle are right angles. So,

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: The opposite sides of a rectangle are parallel and ANSWER: congruent. So, ZY = WX. 39 2x + 3 = x + 4 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. x = 1 Use the value of x to find WX. SOLUTION: WX = 1 + 4 = 5. The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY. ANSWER: 5

15. If PY = 3x – 5 and WP = 2x + 11, find ZP. Then ZP = 4(7) – 9 = 19. SOLUTION: Therefore, ZX = 2(ZP) = 38. The diagonals of a rectangle bisect each other. So, ANSWER: PY = WP. 38 18. If , find The diagonals of a rectangle are congruent and . bisect each other. So, is an isosceles triangle. SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 43

16. If , find . ANSWER: SOLUTION: 48 All four angles of a rectangle are right angles. So, 19. If , find . SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. Then,

ANSWER: 39 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. Therefore, Since is a transversal of parallel sides and , alternate SOLUTION: interior angles WZX and ZXY are congruent. So, The diagonals of a rectangle are congruent and

bisect each other. So, ZP = PY. = = 46. ANSWER: 46

Then ZP = 4(7) – 9 = 19. PROOF Write a two-column proof. Therefore, ZX = 2(ZP) = 38. 20. Given: ABCD is a rectangle. Prove: ANSWER: 38

18. If , find . SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. ANSWER: Prove: 48 Proof: Statements (Reasons) 19. If , find 1. ABCD is a rectangle. (Given) . 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) SOLUTION:

All four angles of a rectangle are right angles. So, 4. (Refl. Prop.) is a right triangle. Then, 5. (Diagonals of a rectangle are .) 6. (SSS) ANSWER: Proof: Statements (Reasons) Therefore, Since is a 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) transversal of parallel sides and , alternate

interior angles WZX and ZXY are congruent. So, 3. (Opp. sides of a .) = = 46. 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) ANSWER: 6. (SSS) 46

PROOF Write a two-column proof. 21. Given: QTVW is a rectangle. 20. Given: ABCD is a rectangle. Prove:

Prove: SOLUTION: SOLUTION: You need to walk through the proof step by step. You need to walk through the proof step by step. Look over what you are given and what you need to Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. prove. Here, you are given QTVW is a rectangle and You need to prove . Use the properties that you have learned about rectangles to . You need to prove . Use walk through the proof. the properties that you have learned about rectangles to walk through the proof. Given: ABCD is a rectangle.

Prove: Given: QTVW is a rectangle. Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .)

4. (Refl. Prop.) Prove: 5. (Diagonals of a rectangle are .) Proof: 6. (SSS) Statements (Reasons) 1. QTVW is a rectangle; .(Given) ANSWER: 2. QTVW is a parallelogram. (Def. of rectangle) Proof:

Statements (Reasons) 3. (Opp sides of a .) 1. ABCD is a rectangle. (Given) 4. are right angles.(Def. of rectangle) 2. ABCD is a parallelogram. (Def. of rectangle) 5. (All rt .) 3. (Opp. sides of a .) 6. QR = ST (Def. of segs.) 4. (Refl. Prop.) 7. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 8. RS = RS (Def. of segs.) 6. (SSS) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 21. Given: QTVW is a rectangle.

12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) Prove: 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) SOLUTION: 3. (Opp sides of a .) You need to walk through the proof step by step. Look over what you are given and what you need to 4. are right angles.(Def. of rectangle) prove. Here, you are given QTVW is a rectangle and 5. (All rt .) . You need to prove . Use 6. QR = ST (Def. of segs.) the properties that you have learned about rectangles 7. (Refl. Prop.) to walk through the proof. 8. RS = RS (Def. of segs.)

9. QR + RS = RS + ST (Add. prop.)

Given: QTVW is a rectangle. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine Prove: whether the figure is a rectangle. Justify your Proof: answer using the indicated formula. Statements (Reasons) 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula 1. QTVW is a rectangle; .(Given) SOLUTION: 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) Use the slope formula to find the slope of the sides of 13. (SAS) the quadrilateral. ANSWER: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) The slopes of each pair of opposite sides are equal. 7. (Refl. Prop.) So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. 8. RS = RS (Def. of segs.) The products of the slopes of the adjacent sides are

9. QR + RS = RS + ST (Add. prop.) –1. So, any two adjacent sides are perpendicular to

10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) each other. That is, all the four angles are right

11. QS = RT (Subst.) angles. Therefore, WXYZ is a rectangle. 12. (Def. of segs.) 13. (SAS) ANSWER:

COORDINATE GEOMETRY Graph each Yes; slope of slope of , slope quadrilateral with the given vertices. Determine of slope of . So WXYZ is a whether the figure is a rectangle. Justify your parallelogram. The product of the slopes of answer using the indicated formula. consecutive sides is –1, so the consecutive sides are 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula perpendicular and form right angles. Thus, WXYZ is a SOLUTION: rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral. 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle. First, Use the Distance formula to find the lengths of the sides of the quadrilateral. ANSWER:

Yes; slope of slope of , slope

of slope of . So WXYZ is a parallelogram. The product of the slopes of so JKLM is a consecutive sides is –1, so the consecutive sides are parallelogram. perpendicular and form right angles. Thus, WXYZ is a A parallelogram is a rectangle if the diagonals are rectangle. congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals. First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: so QRST is a No; so JKLM is a parallelogram. parallelogram; so A parallelogram is a rectangle if the diagonals are the diagonals are not congruent. Thus, JKLM is not a congruent. Use the Distance Formula to find the

rectangle. lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the First, use the slope formula to find the lengths of the lengths of the diagonals. sides of the quadrilateral.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle. The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle.

ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a 625. -4 RectanglesG(1, 8), H(– 7, 7), J(–6, 1), K(2, 2); Slope Formula parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are SOLUTION: not perpendicular. Thus, GHJK is not a rectangle.

First, use the slope formula to find the lengths of the sides of the quadrilateral. Quadrilateral ABCD is a rectangle. Find each measure if .

26. SOLUTION: The slopes of each pair of opposite sides are equal. All four angles of a rectangle are right angles. So, So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. ANSWER: Therefore, GHJK is not a rectangle. 50 ANSWER: 27. No; slope of slope of and slope of SOLUTION: slope of . So, GHJK is a The measures of angles 2 and 3 are equal as they parallelogram. The product of the slopes of are alternate interior angles. consecutive sides ≠ –1, so the consecutive sides are Since the diagonals of a rectangle are congruent and not perpendicular. Thus, GHJK is not a rectangle. bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: 40

28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. eSolutions Manual - Powered by Cognero Page 9 ANSWER: Quadrilateral ABCD is a rectangle. Find each 40 measure if . 29. SOLUTION: All four angles of a rectangle are right angles. So,

26. The measures of angles 1 and 4 are equal as they SOLUTION: are alternate interior angles. All four angles of a rectangle are right angles. So, Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. ANSWER: Therefore, m∠5 = 180 – (50 + 50) = 80. 50 ANSWER: 80 27. SOLUTION: 30. The measures of angles 2 and 3 are equal as they SOLUTION: are alternate interior angles. All four angles of a rectangle are right angles. So, Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7

and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. The measures of angles 1 and 4 are equal as they ANSWER: are alternate interior angles. 40 Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and 28. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = SOLUTION: m∠4 = 50. The measures of angles 2 and 3 are equal as they are alternate interior angles. ANSWER: Therefore, m∠3 = m∠2 = 40. 50

ANSWER: 31. 40 SOLUTION: 29. The measures of angles 2 and 3 are equal as they are alternate interior angles. SOLUTION: Since the diagonals of a rectangle are congruent and All four angles of a rectangle are right angles. So, bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. Therefore, m∠8 = 180 – (40 + 40) = 100. The measures of angles 1 and 4 are equal as they are alternate interior angles. ANSWER: Therefore, m∠4 = m∠1 = 50. 100 Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. shown. To what length should she cut the metal supports in order for the bookshelf to be square, Therefore, m 5 = 180 – (50 + 50) = 80. ∠ which means that the angles formed by the shelves ANSWER: and the vertical supports are all right angles? Explain your reasoning. 80

30. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they SOLUTION: are alternate interior angles. In order for the bookshelf to be square, the lengths of Therefore, m 4 = m 1 = 50. ∠ ∠ all of the metal supports should be equal. Since we Since the diagonals of a rectangle are congruent and know the length of the bookshelves and the distance bisect each other, the triangle with the angles 4, 5 between them, use the Pythagorean Theorem to find and 6 is an isosceles triangle. Therefore, m∠6 = the lengths of the metal supports. m∠4 = 50. Let x be the length of each support. Express 3 feet as 36 inches. ANSWER:

31. SOLUTION: The measures of angles 2 and 3 are equal as they Therefore, the metal supports should each be 39 are alternate interior angles. inches long or 3 feet 3 inches long. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 ANSWER: and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. 3 ft 3 in.; Sample answer: In order for the bookshelf The sum of the three angles of a triangle is 180. to be square, the lengths of all of the metal supports Therefore, m∠8 = 180 – (40 + 40) = 100. should be equal. Since I knew the length of the bookshelves and the distance between them, I used ANSWER: the Pythagorean Theorem to find the lengths of the 100 metal supports.

32. CCSS MODELING Jody is building a new PROOF Write a two-column proof. bookshelf using wood and metal supports like the one 33. Theorem 6.13 shown. To what length should she cut the metal SOLUTION: supports in order for the bookshelf to be square, You need to walk through the proof step by step. which means that the angles formed by the shelves Look over what you are given and what you need to and the vertical supports are all right angles? Explain prove. Here, you are given WXYZ is a rectangle with your reasoning. diagonals . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: WXYZ is a rectangle with diagonals . Prove:

SOLUTION: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find Proof: the lengths of the metal supports. 1. WXYZ is a rectangle with diagonals . Let x be the length of each support. Express 3 feet (Given) as 36 inches. 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) 5. (All right .) Therefore, the metal supports should each be 39 6. (SAS) inches long or 3 feet 3 inches long. 7. (CPCTC) ANSWER: 3 ft 3 in.; Sample answer: In order for the bookshelf ANSWER: to be square, the lengths of all of the metal supports Given: WXYZ is a rectangle with diagonals should be equal. Since I knew the length of the . bookshelves and the distance between them, I used Prove: the Pythagorean Theorem to find the lengths of the metal supports.

PROOF Write a two-column proof. 33. Theorem 6.13

SOLUTION: Proof: You need to walk through the proof step by step. 1. WXYZ is a rectangle with diagonals . Look over what you are given and what you need to

prove. Here, you are given WXYZ is a rectangle with (Given)

diagonals . You need to prove 2. (Opp. sides of a .)

. Use the properties that you have learned 3. (Refl. Prop.) about rectangles to walk through the proof. 4. are right angles. (Def. of rectangle) Given: WXYZ is a rectangle with diagonals 5. (All right .) . 6. (SAS) Prove: 7. (CPCTC)

34. Theorem 6.14 SOLUTION: You need to walk through the proof step by step. Proof: Look over what you are given and what you need to 1. WXYZ is a rectangle with diagonals . prove. Here, you are given (Given) . You need to 2. (Opp. sides of a .) prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk

3. (Refl. Prop.) through the proof. 4. are right angles. (Def. of rectangle) Given: 5. (All right .) Prove: WXYZ is a rectangle. 6. (SAS) 7. (CPCTC)

ANSWER: Given: WXYZ is a rectangle with diagonals . Proof:

Prove: 1. (Given) 2. (SSS) 3. (CPCTC) 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) Proof: 6. are supplementary. (Cons. 1. WXYZ is a rectangle with diagonals . are suppl.) (Given) 7. (Def of suppl.) 2. (Opp. sides of a .) 8. are right angles. (If

3. (Refl. Prop.) 2 and suppl., each is a rt. .) 4. are right angles. (Def. of 9. are right angles. (If a has 1

rectangle) rt. , it has 4 rt. .) 5. (All right .) 10. WXYZ is a rectangle. (Def. of rectangle) 6. (SAS) ANSWER: 7. (CPCTC) Given: Prove: WXYZ is a rectangle. 34. Theorem 6.14 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given . You need to Proof: prove that WXYZ is a rectangle.. Use the properties 1. (Given) that you have learned about rectangles to walk 2. (SSS) through the proof. 3. (CPCTC)

Given: 4. (Def. of ) Prove: WXYZ is a rectangle. 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If Proof: 2 and suppl., each is a rt. .) 1. (Given) 9. are right angles. (If a has 1 2. (SSS) rt. , it has 4 rt. .) 3. (CPCTC) 10. WXYZ is a rectangle. (Def. of rectangle) 4. (Def. of ) PROOF Write a paragraph proof of each 5. WXYZ is a parallelogram. (If both pairs of opp. statement. sides are , then quad. is .) 35. If a parallelogram has one right angle, then it is a 6. are supplementary. (Cons. rectangle. are suppl.) SOLUTION:

7. (Def of suppl.) You need to walk through the proof step by step. 8. are right angles. (If Look over what you are given and what you need to 2 and suppl., each is a rt. .) prove. Here, you are given a parallelogram with one 9. are right angles. (If a has 1 right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. rt. , it has 4 rt. .) Then use the properties of parallelograms to walk 10. WXYZ is a rectangle. (Def. of rectangle) through the proof. ANSWER: Given: Prove: WXYZ is a rectangle.

ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right Proof: angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. 1. (Given) 2. (SSS) ANSWER: 3. (CPCTC) 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons.

are suppl.) ABCD is a parallelogram, and is a right angle. 7. (Def of suppl.) Since ABCD is a parallelogram and has one right 8. are right angles. (If angle, then it has four right angles. So by the 2 and suppl., each is a rt. .) definition of a rectangle, ABCD is a rectangle. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 36. If a quadrilateral has four right angles, then it is a

10. WXYZ is a rectangle. (Def. of rectangle) rectangle. SOLUTION: PROOF Write a paragraph proof of each You need to walk through the proof step by step. statement. Look over what you are given and what you need to 35. If a parallelogram has one right angle, then it is a prove. Here, you are given a quadrilateral with four rectangle. right angles. You need to prove that it is a rectangle. SOLUTION: Let ABCD be a quadrilateral with four right angles. You need to walk through the proof step by step. Then use the properties of parallelograms and Look over what you are given and what you need to rectangles to walk through the proof. prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

ANSWER: ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle.

ANSWER:

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

CONSTRUCTION ABCD is a parallelogram, and is a right angle. 37. Construct a rectangle using the Since ABCD is a parallelogram and has one right construction for congruent segments and the angle, then it has four right angles. So by the construction for a line perpendicular to another line definition of a rectangle, ABCD is a rectangle. through a point on the line. Justify each step of the construction.

36. If a quadrilateral has four right angles, then it is a SOLUTION: rectangle. Sample answer: Step 1: Graph a line and place points P and Q on the SOLUTION: line. You need to walk through the proof step by step. Look over what you are given and what you need to Step 2: Place the compass on point P. Using the prove. Here, you are given a quadrilateral with four same compass setting, draw an arc to the left and right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. right of P that intersects . Then use the properties of parallelograms and rectangles to walk through the proof. Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Place the compass on each point of intersection and draw arcs that intersect above .

Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of Step 5: Repeat steps 2 and 3 using point Q. opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5. ANSWER: Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that ABCD is a quadrilateral with four right angles. intersects line b. Label the point of intersection S. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 9: Draw .

line.

Step 2: Place the compass on point P. Using the Since and , the measure of same compass setting, draw an arc to the left and angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so right of P that intersects . . The same compass setting was used to Step 3: Open the compass to a setting greater than locate points R and S, so . If one pair of the distance from P to either point of intersection. opposite sides of a quadrilateral is both parallel and Place the compass on each point of intersection and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle.

draw arcs that intersect above . Thus, PRSQ is a rectangle.

Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3. ANSWER:

Step 5: Repeat steps 2 and 3 using point Q. Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that Step 6: Draw line b through point Q and the are perpendicular to the same line are parallel, so intersection of the arcs drawn in step 5. . The same compass setting was used to Step 7: Using any compass setting, put the compass locate points R and S, so . If one pair of at P and draw an arc above it that intersects line a. opposite sides of a quadrilateral is both parallel and Label the point of intersection R. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Step 8: Using the same compass setting, put the Thus, PRSQ is a rectangle. compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw .

38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is Since and , the measure of the regulation size and be sure that it is also a angles P and Q is 90 degrees. Lines that are rectangle using only a tape measure. perpendicular to the same line are parallel, so SOLUTION: . The same compass setting was used to Sample answer: If the diagonals of a parallelogram locate points R and S, so . If one pair of are congruent, then it is a rectangle. He should opposite sides of a quadrilateral is both parallel and measure the diagonals of the end zone and both congruent, then the quadrilateral is a parallelogram. pairs of opposite sides. If both pairs of opposite sides A parallelogram with right angles is a rectangle. are congruent, then the quadrilateral is a Thus, PRSQ is a rectangle. parallelogram. If the diagonals are congruent, then the end zone is a rectangle.

ANSWER: ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the Sample answer: Since and , the diagonals and both pairs of opposite sides are measure of angles P and Q is 90 degrees. Lines that congruent, then the end zone is a rectangle. are perpendicular to the same line are parallel, so ALGEBRA Quadrilateral WXYZ is a rectangle. . The same compass setting was used to locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. 39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 WZ .

Therefore, YW = 5. 38. SPORTS The end zone of a football field is 160 feet ANSWER: wide and 30 feet long. Kyle is responsible for 5 painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: The diagonals of a rectangle are congruent to each SOLUTION: other. So, WY = XZ = 2c. All four angles of a Sample answer: If the diagonals of a parallelogram rectangle are right angles. So, is a right are congruent, then it is a rectangle. He should 2 2 measure the diagonals of the end zone and both triangle. By the Pythagorean Theorem, XZ = XY + 2 pairs of opposite sides. If both pairs of opposite sides ZY . are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then the end zone is a rectangle.

ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the Therefore, WY = 2(5) = 10. diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. ANSWER: 10 ALGEBRA Quadrilateral WXYZ is a rectangle. 41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your reasoning.

Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that Therefore, YW = 5. can be used to prove that it is a rectangle. If the ANSWER: diagonals could be proven to be congruent, or if the 5 angles could be proven to be right angles, then we could prove that the sign was a rectangle.

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. ANSWER: No; sample answer: Both pairs of opposite sides are SOLUTION: congruent, so the sign is a parallelogram, but no The diagonals of a rectangle are congruent to each measure is given that can be used to prove that it is a other. So, WY = XZ = 2c. All four angles of a rectangle. rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XY2 + PROOF Write a coordinate proof of each 2 statement. ZY . 42. The diagonals of a rectangle are congruent.

SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, Therefore, WY = 2(5) = 10. b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what ANSWER: you need to prove. Here, you are given ABCD is a 10 rectangle and you need to prove . Use the 41. SIGNS The sign below is in the foyer of Nyoko’s properties that you have learned about rectangles to school. Based on the dimensions given, can Nyoko walk through the proof. be sure that the sign is a rectangle? Explain your reasoning. Given: ABCD is a rectangle. Prove:

SOLUTION: Proof: Use the Distance Formula to find the lengths of the Both pairs of opposite sides are congruent, so the diagonals. sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle. The diagonals, have the same length, so ANSWER: they are congruent. No; sample answer: Both pairs of opposite sides are ANSWER: congruent, so the sign is a parallelogram, but no Given: ABCD is a rectangle. measure is given that can be used to prove that it is a

rectangle. Prove:

PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b Proof: units. Then the rest of the vertices are A(0, b), B(a, Use the Distance Formula to find b), and C(a, 0). You need to walk through the proof and . step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a have the same length, so they are rectangle and you need to prove . Use the congruent. properties that you have learned about rectangles to walk through the proof. 43. If the diagonals of a parallelogram are congruent, then it is a rectangle. Given: ABCD is a rectangle. SOLUTION: Prove: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that Proof: ABCD is a rectangle. Use the properties that you Use the Distance Formula to find the lengths of the have learned about rectangles to walk through the diagonals. proof.

Given: ABCD and Prove: ABCD is a rectangle. The diagonals, have the same length, so they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove: Proof:

AC = BD. So,

Proof: Use the Distance Formula to find and . have the same length, so they are congruent. Because A and B are different points, 43. If the diagonals of a parallelogram are congruent, a ≠ 0. Then b = 0. The slope of is undefined then it is a rectangle. and the slope of . Thus, . ∠DAB is SOLUTION: a right angle and ABCD is a rectangle. Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let ANSWER: the length of the bases be a units and the height be c Given: ABCD and units. Then the rest of the vertices are B(a, 0), C(b + Prove: ABCD is a rectangle. a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof. Proof:

Given: ABCD and Prove: ABCD is a rectangle.

But AC = BD. So,

Proof:

Because A and B are different points,

a ≠ 0. Then b = 0. The slope of is undefined AC = BD. So, and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.. 44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one Because A and B are different points, parallelogram ABCD, one MNOP, and one WXYZ. a ≠ 0. Then b = 0. The slope of is undefined Draw the two diagonals of each parallelogram and label the intersections R. and the slope of . Thus, . ∠DAB is b. TABULAR Use a protractor to measure the

a right angle and ABCD is a rectangle. appropriate angles and complete the table below. ANSWER: Given: ABCD and c. VERBAL Make a conjecture about the diagonals Prove: ABCD is a rectangle. of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

Proof:

But AC = BD. So,

b. Use a protractor to measure each angle listed in the table.

Because A and B are different points, c. Sample answer: Each of the angles listed in the a ≠ 0. Then b = 0. The slope of is undefined table are right angles. These angles were each and the slope of . Thus, . ∠DAB is formed by diagonals. The diagonals of a a right angle and ABCD is a rectangle.. parallelogram with four congruent sides are perpendicular. 44. MULTIPLE REPRESENTATIONS In the ANSWER: problem, you will explore properties of other special parallelograms. a. Sample answer: a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: b. a. Sample answer:

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

b. Use a protractor to measure each angle listed in the table.

c. Sample answer: Each of the angles listed in the SOLUTION: table are right angles. These angles were each All four angles of a rectangle are right angles. So, formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: The diagonals of a rectangle are congruent and a. Sample answer: bisect each other. So, is an isosceles triangle with So,

The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

ANSWER: c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular. x = 6, y = −10

45. CHALLENGE In rectangle ABCD, 46. CCSS CRITIQUE Parker says that any two Find the congruent acute triangles can be arranged to make a values of x and y. rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

SOLUTION: All four angles of a rectangle are right angles. So, In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right

angle.

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So,

So, Tamika is correct.

ANSWER: The sum of the angles of a triangle is 180. Tamika; Sample answer: When two congruent So, triangles are arranged to form a quadrilateral, two of By the Vertical Angle Theorem, the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. ANSWER: How many rectangles are formed by the intersecting x = 6, y = −10 lines?

When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a SOLUTION: single vertex of a triangle. Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 So, Tamika is correct. rectangles are formed by the intersecting lines.

ANSWER: ANSWER: Tamika; Sample answer: When two congruent 6 triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. 48. OPEN ENDED Write the equations of four lines In order for the quadrilateral to be a rectangle, one of having intersections that form the vertices of a the angles in the congruent triangles has to be a right rectangle. Verify your answer using coordinate angle. geometry. SOLUTION: 47. REASONING In the diagram at the right, lines n, p, Sample answer: q,and r are parallel and lines l and m are parallel. A rectangle has 4 right angles and each pair of How many rectangles are formed by the intersecting opposite sides is parallel and congruent. Lines that lines? run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

x = 0, x = 6, y = 0, y = 4;

SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has Then the rectangles formed are ABXW, ACYW, one right angle, it has four right angles. Therefore, by BCYX, BDZX, CDZY, and ADZW. Therefore, 6 definition, the parallelogram is a rectangle. rectangles are formed by the intersecting lines. ANSWER: ANSWER: Sample answer: x = 0, x = 6, y = 0, 6 y = 4; 48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. The length of is 6 − 0 or 6 units and the length of Selecting y = 0 and x = 0 will form 2 sides of the is 6 – 0 or 6 units. The slope of AB is 0 and the rectangle. The lines y = 4 and x = 6 form the other 2 slope of DC is 0. Since a pair of sides of the sides. quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. x = 0, x = 6, y = 0, y = 4; Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not rectangles. SOLUTION: The length of is 6 − 0 or 6 units and the length of Sample answer: A parallelogram is a quadrilateral in is 6 – 0 or 6 units. The slope of AB is 0 and the which each pair of opposite sides are parallel. By slope of DC is 0. Since a pair of sides of the definition of rectangle, all rectangles have both pairs quadrilateral is both parallel and congruent, by of opposite sides parallel. So rectangles are always Theorem 6.12, the quadrilateral is a parallelogram. parallelograms.

Since is horizontal and is vertical, the lines Parallelograms with right angles are rectangles, so are perpendicular and the measure of the angle they some parallelograms are rectangles, but others with form is 90. By Theorem 6.6, if a parallelogram has non-right angles are not. one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. ANSWER: ANSWER: Sample answer: All rectangles are parallelograms because, by definition, both pairs of opposite sides Sample answer: x = 0, x = 6, y = 0, are parallel. Parallelograms with right angles are y = 4; rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram FGHJ a rectangle?

The length of is 6 − 0 or 6 units and the length of A x = 3, y = 4 is 6 – 0 or 6 units. The slope of AB is 0 and the B x = 4, y = 3 slope of DC is 0. Since a pair of sides of the C x = 7, y = 8 quadrilateral is both parallel and congruent, by D x = 8, y = 7 Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines SOLUTION: are perpendicular and the measure of the angle they The opposite sides of a parallelogram are congruent. form is 90. By Theorem 6.6, if a parallelogram has So, FJ = GH. one right angle, it has four right angles. Therefore, by –3x + 5y = 11 FJ = GH definition, the parallelogram is a rectangle. If the diagonals of a parallelogram are congruent, 49. WRITING IN MATH Explain why all rectangles then it is a rectangle. Since FGHJ is a parallelogram, are parallelograms, but all parallelograms are not the diagonals bisect each other. So, if it is a rectangles. rectangle, then will be an isosceles triangle with That is, 3x + y = 13. SOLUTION:

Sample answer: A parallelogram is a quadrilateral in Add the two equations to eliminate the x-term and which each pair of opposite sides are parallel. By solve for y. definition of rectangle, all rectangles have both pairs –3x + 5y = 11 of opposite sides parallel. So rectangles are always 3x + y = 13 parallelograms. 6y = 24 Add to eliminate x-term.

Parallelograms with right angles are rectangles, so y = 4 Divide each side by 6.

some parallelograms are rectangles, but others with

non-right angles are not. Use the value of y to find the value of x. 3x + 4 = 13 Substitute. ANSWER: 3x = 9 Subtract 4 from each side. Sample answer: All rectangles are parallelograms x = 3 Divide each side by 3. because, by definition, both pairs of opposite sides are parallel. Parallelograms with right angles are So, x = 3, y = 4, the correct choice is A. rectangles, so some parallelograms are rectangles, ANSWER: but others with non-right angles are not. A 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram 51. ALGEBRA A rectangular playground is surrounded FGHJ a rectangle? by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following equations could be used to find r, the shorter side of the playground? F 10r + r = 80 G 4r + 10 = 80 A x = 3, y = 4 H r(r + 10) = 80 B x = 4, y = 3 J 2(r + 10) + 2r = 80 C x = 7, y = 8 SOLUTION: D x = 8, y = 7 The formula to find the perimeter of a rectangle is SOLUTION: given by P = 2(l + w). The perimeter is given to be 8 The opposite sides of a parallelogram are congruent. ft. The width is r and the length is r + 10. So, 2r + 2 So, FJ = GH. (r + 10) = 80. –3x + 5y = 11 FJ = GH Therefore, the correct choice is J.

If the diagonals of a parallelogram are congruent, ANSWER: then it is a rectangle. Since FGHJ is a parallelogram, J the diagonals bisect each other. So, if it is a rectangle, then will be an isosceles triangle 52. SHORT RESPONSE What is the measure of with That is, 3x + y = 13. ?

Add the two equations to eliminate the x-term and solve for y. –3x + 5y = 11 3x + y = 13 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6. SOLUTION: The diagonals of a rectangle are congruent and Use the value of y to find the value of x. bisect each other. So, is an isosceles triangle 3x + 4 = 13 Substitute.

3x = 9 Subtract 4 from each side. with

x = 3 Divide each side by 3. So,

By the Exterior Angle Theorem, So, x = 3, y = 4, the correct choice is A.

ANSWER: ANSWER: A 112

51. ALGEBRA A rectangular playground is surrounded 53. SAT/ACT If p is odd, which of the following must by an 80-foot fence. One side of the playground is 10 also be odd? feet longer than the other. Which of the following A 2p equations could be used to find r, the shorter side of B 2p + 2 the playground? C F 10r + r = 80 G 4r + 10 = 80 D H r(r + 10) = 80 E p + 2 J 2(r + 10) + 2r = 80 SOLUTION: SOLUTION: Analyze each answer choice to determine which The formula to find the perimeter of a rectangle is expression is odd when p is odd. given by P = 2(l + w). The perimeter is given to be 8 Choice A 2p ft. The width is r and the length is r + 10. So, 2r + 2 This is a multiple of 2 so it is even. (r + 10) = 80. Choice B 2p + 2 Therefore, the correct choice is J. This can be rewritten as 2(p + 1) which is also a ANSWER: multiple of 2 so it is even.

J Choice C

52. SHORT RESPONSE What is the measure of If p is odd , is not a whole number. ? Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. The correct choice is E. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and E bisect each other. So, is an isosceles triangle with ALGEBRA Find x and y so that the So, quadrilateral is a parallelogram. By the Exterior Angle Theorem,

ANSWER: 112 54. 53. SAT/ACT If p is odd, which of the following must SOLUTION: also be odd? The opposite sides of a parallelogram are congruent. A 2p B 2p + 2

C In a parallelogram, consecutive interior angles are D supplementary. E p + 2 SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. ANSWER: Choice B 2p + 2

This can be rewritten as 2(p + 1) which is also a x = 2, y = 41 multiple of 2 so it is even.

Choice C If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a 55. multiple of 2 so it is even. Choice E p + 2 SOLUTION: Adding 2 to an odd number results in an odd The opposite sides of a parallelogram are congruent. number. This is odd. The correct choice is E.

ANSWER: The opposite sides of a parallelogram are parallel to E each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and ALGEBRA Find x and y so that the hence they are equal. quadrilateral is a parallelogram.

ANSWER: 54. x = 8, y = 22 SOLUTION: The opposite sides of a parallelogram are congruent.

56. In a parallelogram, consecutive interior angles are supplementary. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 Use the value of y in the second equation and solve ANSWER: for x.

x = 2, y = 41 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: x = 2, y = 7

57. COORDINATE GEOMETRY Find the 55. coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C SOLUTION: (4, –2), and D(–1, –1). The opposite sides of a parallelogram are congruent. SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. The opposite sides of a parallelogram are parallel to Find the midpoint of or . each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal. Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5) ANSWER: x = 8, y = 22 Refer to the figure.

56. SOLUTION:

The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. 58. If , name two congruent angles. Solve the first equation to find the value of y. SOLUTION: 2y = 14 y = 7 Since is an isosceles triangle. So, Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 ANSWER: 4 = 2x 2 = x

ANSWER: 59. If , name two congruent segments. x = 2, y = 7 SOLUTION: In a triangle with two congruent angles, the sides COORDINATE GEOMETRY 57. Find the opposite to the congruent angles are also congruent. coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C So, . (4, –2), and D(–1, –1). ANSWER: SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. 60. If , name two congruent segments. Find the midpoint of or . SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent.

Therefore, the coordinates of the intersection of the So, . diagonals are (2.5, 0.5). ANSWER: ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. SOLUTION: FARMING An X-brace on a rectangular barn The diagonals of a rectangle are congruent and door is both decorative and functional. It helps bisect each other. So, is an isosceles triangle. to prevent the door from warping over time. Then, By the Exterior Angle Theorem, If feet, PS = 7 feet, and

, find each measure. Therefore,

ANSWER: 33.5

4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

1. QR By the Vertical Angle Theorem,

SOLUTION: The opposite sides of a rectangle are parallel and Therefore, congruent. Therefore, QR = PS = 7 ft. ANSWER: ANSWER: 56.5 7 ft ALGEBRA Quadrilateral DEFG is a rectangle. 2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

ANSWER: Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 3. 11 SOLUTION: 6. If , The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. find . Then, SOLUTION: By the Exterior Angle Theorem, All four angles of a rectangle are right angles. So,

Therefore,

ANSWER:

33.5

4. ANSWER: SOLUTION: 51 The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 7. PROOF If ABDE is a rectangle and , Then, prove that . By the Vertical Angle Theorem,

Therefore,

ANSWER: 56.5

ALGEBRA Quadrilateral DEFG is a rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and . You need to prove . Use the

5. If FD = 3x – 7 and EG = x + 5, find EG. properties that you have learned about rectangles to walk through the proof. SOLUTION: The diagonals of a rectangle are congruent to each Given: ABDE is a rectangle; other. So, FD = EG. Prove:

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER:

11 Statements(Reasons) 1. ABDE is a rectangle; . 6. If , (Given) find . 2. ABDE is a parallelogram. (Def. of rectangle) SOLUTION: 3. (Opp. sides of a .) All four angles of a rectangle are right angles. So, 4. are right angles.(Def. of rectangle)

5. (All rt .)

6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; ANSWER: Prove: 51

7. PROOF If ABDE is a rectangle and , prove that .

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle)

SOLUTION: 3. (Opp. sides of a .) You need to walk through the proof step by step. 4. are right angles.(Def. of rectangle) Look over what you are given and what you need to 5. (All rt .) prove. You are given ABDE is a rectangle and 6. (SAS) . You need to prove . Use the 7. (CPCTC) properties that you have learned about rectangles to walk through the proof. COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine Given: ABDE is a rectangle; whether the figure is a rectangle. Justify your answer using the indicated formula. Prove: 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle)

3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) Use the slope formula to find the slope of the sides of 5. (All rt .) the quadrilateral. 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

Statements(Reasons) No; slope of , slope of , slope of 1. ABDE is a rectangle; . (Given) , and slope of . Slope of 2. ABDE is a parallelogram. (Def. of rectangle) slope of , and slope of slope of 3. (Opp. sides of a .) , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

There are no parallel sides, so WXYZ is not a so ABCD is a parallelogram. Therefore, WXYZ is not a rectangle. parallelogram. A parallelogram is a rectangle if the diagonals are ANSWER: congruent. Use the Distance formula to find the lengths of the diagonals. No; slope of , slope of , slope of

, and slope of . Slope of So the diagonals are congruent. Thus, ABCD is a slope of , and slope of slope of rectangle. , so WXYZ is not a parallelogram. Therefore, ANSWER: WXYZ is not a rectangle. Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

First, Use the Distance formula to find the lengths of 10. BC the sides of the quadrilateral. SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: so ABCD is a 2 ft parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the 11. DB lengths of the diagonals. SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 So the diagonals are congruent. Thus, ABCD is a Theorem, DB = BC + DC . rectangle. BC = AD = 2 ANSWER: DC = AB = 6 Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. ANSWER: 6.3 ft

12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, By the Vertical Angle Theorem, AD = 2 feet, and , find each measure. ANSWER: 50

13. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, 10. BC All four angles of a rectangle are right angles. SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: ANSWER: 2 ft 25 CCSS REGULARITY Quadrilateral WXYZ is a 11. DB rectangle. SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6 14. If ZY = 2x + 3 and WX = x + 4, find WX.

SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. ANSWER: 2x + 3 = x + 4 6.3 ft x = 1 Use the value of x to find WX. 12. WX = 1 + 4 = 5. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 5 bisect each other. So, is an isosceles triangle. 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. Then, SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP. By the Vertical Angle Theorem,

The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 50

13. SOLUTION: The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles 43 triangle. Then, 16. If , find All four angles of a rectangle are right angles. . SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 25

CCSS REGULARITY Quadrilateral WXYZ is a rectangle.

ANSWER: 39 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX.

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY. The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 Use the value of x to find WX. Then ZP = 4(7) – 9 = 19. WX = 1 + 4 = 5. Therefore, ZX = 2(ZP) = 38.

ANSWER: ANSWER: 5 38 18. If , find 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. . SOLUTION: SOLUTION: The diagonals of a rectangle bisect each other. So, All four angles of a rectangle are right angles. So, PY = WP.

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

ANSWER: 48 ANSWER: 43 19. If , find . 16. If , find SOLUTION:

. All four angles of a rectangle are right angles. So, SOLUTION: is a right triangle. Then, All four angles of a rectangle are right angles. So,

Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46. ANSWER: 39 ANSWER: 46 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. PROOF Write a two-column proof. SOLUTION: 20. Given: ABCD is a rectangle. The diagonals of a rectangle are congruent and Prove: bisect each other. So, ZP = PY.

Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38. SOLUTION: ANSWER: You need to walk through the proof step by step. 38 Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. 18. If , find You need to prove . Use the . properties that you have learned about rectangles to walk through the proof. SOLUTION: All four angles of a rectangle are right angles. So, Given: ABCD is a rectangle. Prove: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle)

3. (Opp. sides of a .) ANSWER: 4. (Refl. Prop.) 48 5. (Diagonals of a rectangle are .) 6. (SSS) 19. If , find . ANSWER: Proof: SOLUTION: Statements (Reasons) All four angles of a rectangle are right angles. So, 1. ABCD is a rectangle. (Given) is a right triangle. Then, 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS) Therefore, Since is a transversal of parallel sides and , alternate 21. Given: QTVW is a rectangle. interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: 46 PROOF Write a two-column proof. Prove: 20. Given: ABCD is a rectangle. Prove: SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and . You need to prove . Use

the properties that you have learned about rectangles SOLUTION: to walk through the proof. You need to walk through the proof step by step. Look over what you are given and what you need to Given: QTVW is a rectangle. prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove: Prove: Proof: Proof: Statements (Reasons) Statements (Reasons)

1. ABCD is a rectangle. (Given) 1. QTVW is a rectangle; .(Given) 2. ABCD is a parallelogram. (Def. of rectangle) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 3. (Opp sides of a .)

4. (Refl. Prop.) 4. are right angles.(Def. of rectangle)

5. (Diagonals of a rectangle are .) 5. (All rt .)

6. (SSS) 6. QR = ST (Def. of segs.) ANSWER: 7. (Refl. Prop.) Proof: 8. RS = RS (Def. of segs.) Statements (Reasons) 9. QR + RS = RS + ST (Add. prop.) 1. ABCD is a rectangle. (Given) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 2. ABCD is a parallelogram. (Def. of rectangle) 11. QS = RT (Subst.) 3. (Opp. sides of a .) 12. (Def. of segs.) 4. (Refl. Prop.) 13. (SAS) 5. (Diagonals of a rectangle are .) ANSWER: 6. (SSS) Proof: Statements (Reasons) 21. Given: QTVW is a rectangle. 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) Prove: 7. (Refl. Prop.) SOLUTION: 8. RS = RS (Def. of segs.) You need to walk through the proof step by step. 9. QR + RS = RS + ST (Add. prop.) Look over what you are given and what you need to 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) prove. Here, you are given QTVW is a rectangle and 11. QS = RT (Subst.) . You need to prove . Use 12. (Def. of segs.) the properties that you have learned about rectangles 13. (SAS) to walk through the proof. COORDINATE GEOMETRY Graph each Given: QTVW is a rectangle. quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION:

7. (Refl. Prop.) the quadrilateral. 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) The slopes of each pair of opposite sides are equal. 1. QTVW is a rectangle; .(Given) So, the two pairs of opposite sides are parallel. 2. QTVW is a parallelogram. (Def. of rectangle) Therefore, the quadrilateral WXYZ is a parallelogram. 3. (Opp sides of a .) The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to 4. are right angles.(Def. of rectangle) each other. That is, all the four angles are right 5. (All rt .) angles. Therefore, WXYZ is a rectangle. 6. QR = ST (Def. of segs.) ANSWER: 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) Yes; slope of slope of , slope 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) of slope of . So WXYZ is a 11. QS = RT (Subst.) parallelogram. The product of the slopes of 12. (Def. of segs.) consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a

13. (SAS) rectangle. COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION:

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are So the diagonals are not congruent. Thus, JKLM is perpendicular and form right angles. Thus, WXYZ is a not a rectangle. rectangle. ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals. so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the So the diagonals are not congruent. Thus, JKLM is lengths of the diagonals. not a rectangle. ANSWER: No; so JKLM is a So the diagonals are congruent. Thus, QRST is a rectangle. parallelogram; so the diagonals are not congruent. Thus, JKLM is not a ANSWER: rectangle. Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION: 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

First, use the slope formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle. The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. ANSWER: Therefore, the quadrilateral GHJK is a parallelogram. Yes; so QRST is a None of the adjacent sides have slopes whose parallelogram. so the diagonals are product is –1. So, the angles are not right angles. congruent. QRST is a rectangle. Therefore, GHJK is not a rectangle. ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

First, use the slope formula to find the lengths of the sides of the quadrilateral. 26. SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 50 The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a 27. parallelogram. SOLUTION: None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. The measures of angles 2 and 3 are equal as they Therefore, GHJK is not a rectangle. are alternate interior angles. Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. No; slope of slope of and slope of Therefore, m∠7 = m∠2 = 40. slope of . So, GHJK is a ANSWER: parallelogram. The product of the slopes of 40 consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle. 28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

Quadrilateral ABCD is a rectangle. Find each measure if . The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. 26. The sum of the three angles of a triangle is 180. SOLUTION: Therefore, m∠5 = 180 – (50 + 50) = 80. All four angles of a rectangle are right angles. So, ANSWER: 80

30.

ANSWER: SOLUTION: 50 All four angles of a rectangle are right angles. So,

27. SOLUTION: The measures of angles 1 and 4 are equal as they The measures of angles 2 and 3 are equal as they are alternate interior angles. are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 bisect each other, the triangle with the angles 4, 5 and 8 is an isosceles triangle. So, m∠7 = m∠3. and 6 is an isosceles triangle. Therefore, m∠6 = Therefore, m∠7 = m∠2 = 40. m∠4 = 50.

6-4 RectanglesANSWER: ANSWER: 40 50

28. 31. SOLUTION: SOLUTION: The measures of angles 2 and 3 are equal as they The measures of angles 2 and 3 are equal as they are alternate interior angles. are alternate interior angles. Therefore, m∠3 = m∠2 = 40. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 ANSWER: and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. 40 The sum of the three angles of a triangle is 180. Therefore, m∠8 = 180 – (40 + 40) = 100. 29. ANSWER: SOLUTION: 100 All four angles of a rectangle are right angles. So, 32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one shown. To what length should she cut the metal The measures of angles 1 and 4 are equal as they supports in order for the bookshelf to be square, are alternate interior angles. which means that the angles formed by the shelves Therefore, m∠4 = m∠1 = 50. and the vertical supports are all right angles? Explain Since the diagonals of a rectangle are congruent and your reasoning. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: 80

are alternate interior angles. as 36 inches.

Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = m∠4 = 50. Therefore, the metal supports should each be 39 ANSWER: inches long or 3 feet 3 inches long. 50 ANSWER: 31. 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports SOLUTION: should be equal. Since I knew the length of the The measures of angles 2 and 3 are equal as they bookshelves and the distance between them, I used are alternate interior angles. the Pythagorean Theorem to find the lengths of the eSolutionsSinceManual the diagonals- Powered by ofCognero a rectangle are congruent and metal supports. Page 10 bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. PROOF Write a two-column proof. The sum of the three angles of a triangle is 180. 33. Theorem 6.13 Therefore, m∠8 = 180 – (40 + 40) = 100. SOLUTION: You need to walk through the proof step by step. ANSWER: Look over what you are given and what you need to 100 prove. Here, you are given WXYZ is a rectangle with 32. CCSS MODELING Jody is building a new diagonals . You need to prove bookshelf using wood and metal supports like the one . Use the properties that you have learned shown. To what length should she cut the metal about rectangles to walk through the proof. supports in order for the bookshelf to be square, which means that the angles formed by the shelves Given: WXYZ is a rectangle with diagonals and the vertical supports are all right angles? Explain your reasoning. . Prove:

Proof: 1. WXYZ is a rectangle with diagonals . (Given) SOLUTION:

In order for the bookshelf to be square, the lengths of 2. (Opp. sides of a .) all of the metal supports should be equal. Since we 3. (Refl. Prop.) know the length of the bookshelves and the distance 4. are right angles. (Def. of between them, use the Pythagorean Theorem to find rectangle) the lengths of the metal supports. 5. (All right .) Let x be the length of each support. Express 3 feet as 36 inches. 6. (SAS) 7. (CPCTC)

ANSWER: Given: WXYZ is a rectangle with diagonals . Therefore, the metal supports should each be 39 Prove: inches long or 3 feet 3 inches long.

PROOF Write a two-column proof. 2. (Opp. sides of a .) 33. Theorem 6.13 3. (Refl. Prop.) SOLUTION: 4. are right angles. (Def. of You need to walk through the proof step by step. rectangle) Look over what you are given and what you need to 5. (All right .) prove. Here, you are given WXYZ is a rectangle with 6. (SAS) diagonals . You need to prove 7. (CPCTC) . Use the properties that you have learned about rectangles to walk through the proof. 34. Theorem 6.14

Given: WXYZ is a rectangle with diagonals SOLUTION: You need to walk through the proof step by step. . Look over what you are given and what you need to Prove: prove. Here, you are given . You need to prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk through the proof.

Proof: Given: 1. WXYZ is a rectangle with diagonals . Prove: WXYZ is a rectangle. (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) 5. (All right .) Proof: 6. (SAS) 1. (Given) 7. (CPCTC) 2. (SSS) 3. (CPCTC) ANSWER: 4. (Def. of ) Given: WXYZ is a rectangle with diagonals 5. WXYZ is a parallelogram. (If both pairs of opp. . sides are , then quad. is .) Prove: 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .)

Proof: 9. are right angles. (If a has 1

1. WXYZ is a rectangle with diagonals . rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) (Given) 2. (Opp. sides of a .) ANSWER: 3. (Refl. Prop.) Given: 4. are right angles. (Def. of Prove: WXYZ is a rectangle. rectangle) 5. (All right .) 6. (SAS) 7. (CPCTC)

Proof: 34. Theorem 6.14 1. (Given) SOLUTION: 2. (SSS) You need to walk through the proof step by step. 3. (CPCTC) Look over what you are given and what you need to 4. (Def. of ) prove. Here, you are given 5. WXYZ is a parallelogram. (If both pairs of opp. . You need to sides are , then quad. is .) prove that WXYZ is a rectangle.. Use the properties 6. are supplementary. (Cons. that you have learned about rectangles to walk are suppl.) through the proof. 7. (Def of suppl.) Given: 8. are right angles. (If Prove: WXYZ is a rectangle. 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each statement. Proof: 35. If a parallelogram has one right angle, then it is a 1. (Given) rectangle. 2. (SSS) SOLUTION: 3. (CPCTC) You need to walk through the proof step by step. 4. (Def. of ) Look over what you are given and what you need to 5. WXYZ is a parallelogram. (If both pairs of opp. prove. Here, you are given a parallelogram with one sides are , then quad. is .) right angle. You need to prove that it is a rectangle. 6. are supplementary. (Cons. Let ABCD be a parallelogram with one right angle. are suppl.) Then use the properties of parallelograms to walk through the proof.

7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

ANSWER: ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right Given: angle, then it has four right angles. So by the

Prove: WXYZ is a rectangle. definition of a rectangle, ABCD is a rectangle.

ANSWER:

Proof: 1. (Given)

statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one ABCD is a quadrilateral with four right angles. right angle. You need to prove that it is a rectangle. ABCD is a parallelogram because both pairs of Let ABCD be a parallelogram with one right angle. opposite angles are congruent. By definition of a Then use the properties of parallelograms to walk rectangle, ABCD is a rectangle. through the proof. ANSWER:

ABCD is a parallelogram, and is a right angle. Step 1: Graph a line and place points P and Q on the Since ABCD is a parallelogram and has one right line. angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and 36. If a quadrilateral has four right angles, then it is a rectangle. right of P that intersects .

SOLUTION: Step 3: Open the compass to a setting greater than You need to walk through the proof step by step. the distance from P to either point of intersection. Look over what you are given and what you need to Place the compass on each point of intersection and prove. Here, you are given a quadrilateral with four draw arcs that intersect above . right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and Step 4: Draw line a through point P and the rectangles to walk through the proof. intersection of the arcs drawn in step 3.

Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 9: Draw .

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. Since and , the measure of angles P and Q is 90 degrees. Lines that are SOLUTION: perpendicular to the same line are parallel, so Sample answer: . The same compass setting was used to Step 1: Graph a line and place points P and Q on the line. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 2: Place the compass on point P. Using the A parallelogram with right angles is a rectangle. same compass setting, draw an arc to the left and Thus, PRSQ is a rectangle. right of P that intersects .

Step 3: Open the compass to a setting greater than ANSWER: the distance from P to either point of intersection. Place the compass on each point of intersection and Sample answer: Since and , the draw arcs that intersect above . measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 4: Draw line a through point P and the . The same compass setting was used to intersection of the arcs drawn in step 3. locate points R and S, so . If one pair of Step 5: Repeat steps 2 and 3 using point Q. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Step 6: Draw line b through point Q and the Thus, PRSQ is a rectangle. intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw . 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both Since and , the measure of pairs of opposite sides. If both pairs of opposite sides angles P and Q is 90 degrees. Lines that are are congruent, then the quadrilateral is a perpendicular to the same line are parallel, so parallelogram. If the diagonals are congruent, then the end zone is a rectangle. . The same compass setting was used to locate points R and S, so . If one pair of ANSWER: opposite sides of a quadrilateral is both parallel and Sample answer: He should measure the diagonals of congruent, then the quadrilateral is a parallelogram. the end zone and both pairs of opposite sides. If the A parallelogram with right angles is a rectangle. diagonals and both pairs of opposite sides are Thus, PRSQ is a rectangle. congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle. ANSWER:

Therefore, YW = 5.

ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: The diagonals of a rectangle are congruent to each 38. SPORTS The end zone of a football field is 160 feet other. So, WY = XZ = 2c. All four angles of a wide and 30 feet long. Kyle is responsible for rectangle are right angles. So, is a right painting the field. He has finished the end zone. triangle. By the Pythagorean Theorem, XZ2 = XY2 + Explain how Kyle can confirm that the end zone is 2 the regulation size and be sure that it is also a ZY . rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides Therefore, WY = 2(5) = 10. are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then ANSWER: the end zone is a rectangle. 10

ANSWER: 41. SIGNS The sign below is in the foyer of Nyoko’s Sample answer: He should measure the diagonals of school. Based on the dimensions given, can Nyoko the end zone and both pairs of opposite sides. If the be sure that the sign is a rectangle? Explain your diagonals and both pairs of opposite sides are reasoning. congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: SOLUTION:

The diagonals of a rectangle are congruent to each Both pairs of opposite sides are congruent, so the other. So, YW = XZ = b. All four angles of a sign is a parallelogram, but no measure is given that rectangle are right angles. So, is a right can be used to prove that it is a rectangle. If the triangle. By the Pythagorean Theorem, XZ2 = XW 2 + diagonals could be proven to be congruent, or if the 2 angles could be proven to be right angles, then we WZ . could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are Therefore, YW = 5. congruent, so the sign is a parallelogram, but no ANSWER: measure is given that can be used to prove that it is a rectangle. 5 PROOF Write a coordinate proof of each 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. statement. 42. The diagonals of a rectangle are congruent. SOLUTION: The diagonals of a rectangle are congruent to each SOLUTION: other. So, WY = XZ = 2c. All four angles of a Begin by positioning quadrilateral ABCD on a rectangle are right angles. So, is a right coordinate plane. Place vertex D at the origin. Let 2 2 the length of the bases be a units and the height be b triangle. By the Pythagorean Theorem, XZ = XY + units. Then the rest of the vertices are A(0, b), B(a, 2 ZY . b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Therefore, WY = 2(5) = 10. Given: ABCD is a rectangle. ANSWER: Prove: 10

41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your reasoning.

Proof: Use the Distance Formula to find the lengths of the diagonals.

SOLUTION: The diagonals, have the same length, so

they are congruent. Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that ANSWER: can be used to prove that it is a rectangle. If the Given: ABCD is a rectangle. diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we Prove: could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

PROOF Write a coordinate proof of each Proof: statement. Use the Distance Formula to find 42. The diagonals of a rectangle are congruent. and . SOLUTION: have the same length, so they are Begin by positioning quadrilateral ABCD on a congruent. coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b 43. If the diagonals of a parallelogram are congruent, units. Then the rest of the vertices are A(0, b), B(a, then it is a rectangle. b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what SOLUTION: you need to prove. Here, you are given ABCD is a Begin by positioning parallelogram ABCD on a rectangle and you need to prove . Use the coordinate plane. Place vertex A at the origin. Let properties that you have learned about rectangles to the length of the bases be a units and the height be c walk through the proof. units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the Given: ABCD is a rectangle. proof step by step. Look over what you are given Prove: and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Prove: ABCD is a rectangle.

Proof: Use the Distance Formula to find the lengths of the diagonals.

Proof: The diagonals, have the same length, so they are congruent.

ANSWER:

Given: ABCD is a rectangle. AC = BD. So,

Prove:

Because A and B are different points, Proof:

Use the Distance Formula to find a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is and . a right angle and ABCD is a rectangle. have the same length, so they are congruent. ANSWER: Given: ABCD and 43. If the diagonals of a parallelogram are congruent, Prove: ABCD is a rectangle. then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the Proof: proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the But AC = BD. So, proof.

Given: ABCD and Prove: ABCD is a rectangle.

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is Proof: a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. AC = BD. So, a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

Because A and B are different points, c. VERBAL Make a conjecture about the diagonals

a ≠ 0. Then b = 0. The slope of is undefined of a parallelogram with four congruent sides. and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle. SOLUTION: a. Sample answer: ANSWER: Given: ABCD and Prove: ABCD is a rectangle.

Proof:

b. Use a protractor to measure each angle listed in the table. But AC = BD. So,

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: Because A and B are different points, a. Sample answer: a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

SOLUTION: All four angles of a rectangle are right angles. So, b. Use a protractor to measure each angle listed in the table.

The diagonals of a rectangle are congruent and c. Sample answer: Each of the angles listed in the bisect each other. So, is an isosceles triangle table are right angles. These angles were each with formed by diagonals. The diagonals of a So, parallelogram with four congruent sides are perpendicular.

ANSWER: a. Sample answer: The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two b. congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is c. Sample answer: The diagonals of a parallelogram either of them correct? Explain your reasoning. with four congruent sides are perpendicular. SOLUTION: 45. CHALLENGE In rectangle ABCD, When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a Find the single vertex of a triangle. values of x and y.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

SOLUTION: All four angles of a rectangle are right angles. So,

So, Tamika is correct. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle ANSWER: with Tamika; Sample answer: When two congruent So, triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

The sum of the angles of a triangle is 180. 47. REASONING In the diagram at the right, lines n, p, So, q,and r are parallel and lines l and m are parallel. By the Vertical Angle Theorem, How many rectangles are formed by the intersecting lines?

ANSWER: x = 6, y = −10

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right Then the rectangles formed are ABXW, ACYW, angle. BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a

So, Tamika is correct. rectangle. Verify your answer using coordinate geometry. ANSWER: SOLUTION: Tamika; Sample answer: When two congruent Sample answer: triangles are arranged to form a quadrilateral, two of A rectangle has 4 right angles and each pair of the angles are formed by a single vertex of a triangle. opposite sides is parallel and congruent. Lines that In order for the quadrilateral to be a rectangle, one of run parallel to the x-axis will be parallel to each other the angles in the congruent triangles has to be a right and perpendicular to lines parallel to the y-axis. angle. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 47. REASONING In the diagram at the right, lines n, p, sides. q,and r are parallel and lines l and m are parallel.

How many rectangles are formed by the intersecting lines? x = 0, x = 6, y = 0, y = 4;

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate The length of is 6 − 0 or 6 units and the length of geometry. is 6 – 0 or 6 units. The slope of AB is 0 and the SOLUTION: slope of DC is 0. Since a pair of sides of the Sample answer: quadrilateral is both parallel and congruent, by A rectangle has 4 right angles and each pair of Theorem 6.12, the quadrilateral is a parallelogram. opposite sides is parallel and congruent. Lines that Since is horizontal and is vertical, the lines run parallel to the x-axis will be parallel to each other are perpendicular and the measure of the angle they and perpendicular to lines parallel to the y-axis. form is 90. By Theorem 6.6, if a parallelogram has Selecting y = 0 and x = 0 will form 2 sides of the one right angle, it has four right angles. Therefore, by rectangle. The lines y = 4 and x = 6 form the other 2 definition, the parallelogram is a rectangle. sides. 49. WRITING IN MATH Explain why all rectangles x = 0, x = 6, y = 0, y = 4; are parallelograms, but all parallelograms are not rectangles. SOLUTION: Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms.

Parallelograms with right angles are rectangles, so The length of is 6 − 0 or 6 units and the length of some parallelograms are rectangles, but others with is 6 – 0 or 6 units. The slope of AB is 0 and the non-right angles are not. slope of DC is 0. Since a pair of sides of the ANSWER: quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Sample answer: All rectangles are parallelograms because, by definition, both pairs of opposite sides Since is horizontal and is vertical, the lines are parallel. Parallelograms with right angles are are perpendicular and the measure of the angle they rectangles, so some parallelograms are rectangles, form is 90. By Theorem 6.6, if a parallelogram has but others with non-right angles are not. one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram ANSWER: FGHJ a rectangle? Sample answer: x = 0, x = 6, y = 0, y = 4;

A x = 3, y = 4 B x = 4, y = 3 C x = 7, y = 8 D x = 8, y = 7 SOLUTION:

The opposite sides of a parallelogram are congruent. The length of is 6 − 0 or 6 units and the length of So, FJ = GH. is 6 – 0 or 6 units. The slope of AB is 0 and the –3x + 5y = 11 FJ = GH slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by If the diagonals of a parallelogram are congruent, Theorem 6.12, the quadrilateral is a parallelogram. then it is a rectangle. Since FGHJ is a parallelogram, Since is horizontal and is vertical, the lines the diagonals bisect each other. So, if it is a are perpendicular and the measure of the angle they rectangle, then will be an isosceles triangle form is 90. By Theorem 6.6, if a parallelogram has with That is, 3x + y = 13. one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. Add the two equations to eliminate the x-term and solve for y. 49. WRITING IN MATH Explain why all rectangles –3x + 5y = 11 are parallelograms, but all parallelograms are not 3x + y = 13 rectangles. 6y = 24 Add to eliminate x-term. SOLUTION: y = 4 Divide each side by 6.

Sample answer: A parallelogram is a quadrilateral in

which each pair of opposite sides are parallel. By Use the value of y to find the value of x.

definition of rectangle, all rectangles have both pairs 3x + 4 = 13 Substitute.

of opposite sides parallel. So rectangles are always 3x = 9 Subtract 4 from each side. parallelograms. x = 3 Divide each side by 3.

Parallelograms with right angles are rectangles, so So, x = 3, y = 4, the correct choice is A. some parallelograms are rectangles, but others with ANSWER: non-right angles are not. A ANSWER: Sample answer: All rectangles are parallelograms 51. ALGEBRA A rectangular playground is surrounded because, by definition, both pairs of opposite sides by an 80-foot fence. One side of the playground is 10 are parallel. Parallelograms with right angles are feet longer than the other. Which of the following rectangles, so some parallelograms are rectangles, equations could be used to find r, the shorter side of but others with non-right angles are not. the playground? F 10r + r = 80 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = G 4r + 10 = 80 13, what values of x and y make parallelogram H r(r + 10) = 80 FGHJ a rectangle? J 2(r + 10) + 2r = 80 SOLUTION: The formula to find the perimeter of a rectangle is given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 A x = 3, y = 4 (r + 10) = 80. B x = 4, y = 3 Therefore, the correct choice is J. C x = 7, y = 8 ANSWER: D x = 8, y = 7 J SOLUTION: The opposite sides of a parallelogram are congruent. 52. SHORT RESPONSE What is the measure of So, FJ = GH. ? –3x + 5y = 11 FJ = GH

If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a rectangle, then will be an isosceles triangle

with That is, 3x + y = 13. SOLUTION:

The diagonals of a rectangle are congruent and Add the two equations to eliminate the x-term and solve for y. bisect each other. So, is an isosceles triangle –3x + 5y = 11 with 3x + y = 13 So, 6y = 24 Add to eliminate x-term. By the Exterior Angle Theorem, y = 4 Divide each side by 6.

Use the value of y to find the value of x. ANSWER: 3x + 4 = 13 Substitute. 112 3x = 9 Subtract 4 from each side. SAT/ACT x = 3 Divide each side by 3. 53. If p is odd, which of the following must

also be odd? So, x = 3, y = 4, the correct choice is A. A 2p B 2p + 2 ANSWER: C A D ALGEBRA 51. A rectangular playground is surrounded E p + 2 by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following SOLUTION: equations could be used to find r, the shorter side of Analyze each answer choice to determine which the playground? expression is odd when p is odd. F 10r + r = 80 Choice A 2p G 4r + 10 = 80 This is a multiple of 2 so it is even. H r(r + 10) = 80 Choice B 2p + 2 J 2(r + 10) + 2r = 80 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even. SOLUTION:

The formula to find the perimeter of a rectangle is Choice C given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 If p is odd , is not a whole number. (r + 10) = 80. Choice D 2p – 2 Therefore, the correct choice is J. This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. ANSWER: Choice E p + 2 J Adding 2 to an odd number results in an odd number. This is odd. SHORT RESPONSE 52. What is the measure of The correct choice is E. ? ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle 54. with So, SOLUTION: By the Exterior Angle Theorem, The opposite sides of a parallelogram are congruent.

ANSWER:

112 In a parallelogram, consecutive interior angles are

53. SAT/ACT If p is odd, which of the following must supplementary. also be odd? A 2p B 2p + 2

D ANSWER: E p + 2 x = 2, y = 41 SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a 55. multiple of 2 so it is even. SOLUTION:

Choice C The opposite sides of a parallelogram are congruent.

If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a The opposite sides of a parallelogram are parallel to multiple of 2 so it is even. each other. So, the angles with the measures (3y + Choice E p + 2 3)° and (4y – 19)° are alternate interior angles and Adding 2 to an odd number results in an odd hence they are equal. number. This is odd. The correct choice is E.

ANSWER: E ANSWER: ALGEBRA Find x and y so that the x = 8, y = 22 quadrilateral is a parallelogram.

54. 56. SOLUTION: SOLUTION: The opposite sides of a parallelogram are congruent. The diagonals of a parallelogram bisect each other.

So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 In a parallelogram, consecutive interior angles are Use the value of y in the second equation and solve supplementary. for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: x = 2, y = 7 ANSWER: 57. COORDINATE GEOMETRY Find the x = 2, y = 41 coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. 55. Find the midpoint of or . SOLUTION: The opposite sides of a parallelogram are congruent.

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: x = 8, y = 22

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So, 56.

SOLUTION: The diagonals of a parallelogram bisect each other. ANSWER: So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14

y = 7 59. If , name two congruent segments. Use the value of y in the second equation and solve SOLUTION: for x. In a triangle with two congruent angles, the sides 7 + 3 = 2x + 6 opposite to the congruent angles are also congruent. 4 = 2x 2 = x So, .

ANSWER: ANSWER: x = 2, y = 7

57. COORDINATE GEOMETRY Find the 60. If , name two congruent segments. coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C SOLUTION: (4, –2), and D(–1, –1). In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. SOLUTION: So, . The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. ANSWER: Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem, FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. Therefore,

If feet, PS = 7 feet, and ANSWER: , find each measure. 33.5

4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore,

1. QR ANSWER: SOLUTION: 56.5 The opposite sides of a rectangle are parallel and ALGEBRA Quadrilateral DEFG is a rectangle. congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: SOLUTION: The diagonals of a rectangle bisect each other. So, The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: ANSWER: 11

Therefore,

ANSWER: ANSWER: 51 33.5

4. 7. PROOF If ABDE is a rectangle and , prove that . SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Given: ABDE is a rectangle;

5. If FD = 3x – 7 and EG = x + 5, find EG. Prove: SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

ANSWER:

Given: ABDE is a rectangle; Prove:

ANSWER: 51

PROOF 7. If ABDE is a rectangle and , Statements(Reasons)

prove that . 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .)

6. (SAS) SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. Look over what you are given and what you need to COORDINATE GEOMETRY Graph each prove. You are given ABDE is a rectangle and quadrilateral with the given vertices. Determine . You need to prove . Use the whether the figure is a rectangle. Justify your properties that you have learned about rectangles to answer using the indicated formula. walk through the proof. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION: Given: ABDE is a rectangle; Prove:

Statements(Reasons)

1. ABDE is a rectangle; . (Given) Use the slope formula to find the slope of the sides of 2. ABDE is a parallelogram. (Def. of the quadrilateral. rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER:

Given: ABDE is a rectangle; There are no parallel sides, so WXYZ is not a Prove: parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of

Statements(Reasons) slope of , and slope of slope of 1. ABDE is a rectangle; . , so WXYZ is not a parallelogram. Therefore,

(Given) WXYZ is not a rectangle. 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION: SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral. First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

ANSWER: So the diagonals are congruent. Thus, ABCD is a No; slope of , slope of , slope of rectangle. ANSWER: , and slope of . Slope of Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a slope of , and slope of slope of parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

2 ft

11. DB so ABCD is a SOLUTION: parallelogram. All four angles of a rectangle are right angles. So, A parallelogram is a rectangle if the diagonals are is a right triangle. By the Pythagorean congruent. Use the Distance formula to find the 2 2 2 Theorem, DB = BC + DC . lengths of the diagonals.

BC = AD = 2 DC = AB = 6

So the diagonals are congruent. Thus, ABCD is a rectangle.

By the Vertical Angle Theorem,

ANSWER: FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, 50 AD = 2 feet, and , find each 13. measure. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, All four angles of a rectangle are right angles.

10. BC ANSWER: SOLUTION: 25 The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. CCSS REGULARITY Quadrilateral WXYZ is a ANSWER: rectangle. 2 ft

11. DB SOLUTION:

All four angles of a rectangle are right angles. So, 14. If ZY = 2x + 3 and WX = x + 4, find WX. is a right triangle. By the Pythagorean 2 2 2 SOLUTION: Theorem, DB = BC + DC . The opposite sides of a rectangle are parallel and BC = AD = 2 congruent. So, ZY = WX. DC = AB = 6 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5.

ANSWER: ANSWER: 6.3 ft 5 12. 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, is an isosceles The diagonals of a rectangle bisect each other. So, triangle. PY = WP. Then,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. By the Vertical Angle Theorem,

ANSWER: 50 ANSWER: 43 13. SOLUTION: 16. If , find The diagonals of a rectangle are congruent and . bisect each other. So, is an isosceles SOLUTION: triangle. All four angles of a rectangle are right angles. So, Then,

All four angles of a rectangle are right angles.

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION:

The opposite sides of a rectangle are parallel and Then ZP = 4(7) – 9 = 19.

congruent. So, ZY = WX. Therefore, ZX = 2(ZP) = 38. 2x + 3 = x + 4 ANSWER:

x = 1 38 Use the value of x to find WX. WX = 1 + 4 = 5. 18. If , find ANSWER: . 5 SOLUTION: All four angles of a rectangle are right angles. So, 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP.

The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 48

19. If , find . SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 43 is a right triangle. Then,

16. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Therefore, Since is a

transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46. ANSWER: 46

ANSWER: PROOF Write a two-column proof. 39 20. Given: ABCD is a rectangle. Prove: 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY.

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to Then ZP = 4(7) – 9 = 19. prove. Here, you are given ABCD is a rectangle.

Therefore, ZX = 2(ZP) = 38. You need to prove . Use the ANSWER: properties that you have learned about rectangles to walk through the proof. 38

ANSWER: ANSWER: 48 Proof: Statements (Reasons) 19. If , find 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) . 3. (Opp. sides of a .) SOLUTION: 4. (Refl. Prop.) All four angles of a rectangle are right angles. So, is a right triangle. Then, 5. (Diagonals of a rectangle are .) 6. (SSS)

21. Given: QTVW is a rectangle.

Given: QTVW is a rectangle.

Proof: 3. (Opp sides of a .) Statements (Reasons) 1. ABCD is a rectangle. (Given) 4. are right angles.(Def. of rectangle) 2. ABCD is a parallelogram. (Def. of rectangle) 5. (All rt .) 3. (Opp. sides of a .) 6. QR = ST (Def. of segs.) 4. (Refl. Prop.) 7. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 8. RS = RS (Def. of segs.) 6. (SSS) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) ANSWER: 11. QS = RT (Subst.) Proof: 12. (Def. of segs.) Statements (Reasons) 13. (SAS) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) ANSWER: 3. (Opp. sides of a .) Proof:

4. (Refl. Prop.) Statements (Reasons)

5. (Diagonals of a rectangle are .) 1. QTVW is a rectangle; .(Given)

6. (SSS) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .)

21. Given: QTVW is a rectangle. 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) Prove: 11. QS = RT (Subst.) SOLUTION: 12. (Def. of segs.) You need to walk through the proof step by step. 13. (SAS) Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and COORDINATE GEOMETRY Graph each . You need to prove . Use quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your the properties that you have learned about rectangles answer using the indicated formula. to walk through the proof. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula

Given: QTVW is a rectangle. SOLUTION:

Prove: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) Use the slope formula to find the slope of the sides of 4. are right angles.(Def. of rectangle) the quadrilateral.

5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS) The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. ANSWER: Therefore, the quadrilateral WXYZ is a parallelogram. Proof: The products of the slopes of the adjacent sides are Statements (Reasons) –1. So, any two adjacent sides are perpendicular to 1. QTVW is a rectangle; .(Given) each other. That is, all the four angles are right 2. QTVW is a parallelogram. (Def. of rectangle) angles. Therefore, WXYZ is a rectangle. 3. (Opp sides of a .) ANSWER: 4. are right angles.(Def. of rectangle) Yes; slope of slope of , slope 5. (All rt .) 6. QR = ST (Def. of segs.) of slope of . So WXYZ is a 7. (Refl. Prop.) parallelogram. The product of the slopes of 8. RS = RS (Def. of segs.) consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a 9. QR + RS = RS + ST (Add. prop.) rectangle. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION: SOLUTION:

First, use the Distance Formula to find the lengths of First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

the sides of the quadrilateral.

the diagonals are not congruent. Thus, JKLM is not a congruent. QRST is a rectangle. rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula

SOLUTION: 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, use the slope formula to find the lengths of the First, use the Distance Formula to find the lengths of sides of the quadrilateral. the sides of the quadrilateral.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

26. SOLUTION: First, use the slope formula to find the lengths of the All four angles of a rectangle are right angles. So, sides of the quadrilateral.

ANSWER: 50

27. SOLUTION: The slopes of each pair of opposite sides are equal. The measures of angles 2 and 3 are equal as they So, the two pairs of opposite sides are parallel. are alternate interior angles. Therefore, the quadrilateral GHJK is a Since the diagonals of a rectangle are congruent and parallelogram. bisect each other, the triangle with the angles 3, 7 None of the adjacent sides have slopes whose and 8 is an isosceles triangle. So, m∠7 = m∠3. product is –1. So, the angles are not right angles. Therefore, m∠7 = m∠2 = 40. Therefore, GHJK is not a rectangle. ANSWER: ANSWER: 40 No; slope of slope of and slope of 28. slope of . So, GHJK is a SOLUTION: parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are The measures of angles 2 and 3 are equal as they

not perpendicular. Thus, GHJK is not a rectangle. are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 26. 80 SOLUTION: 30. All four angles of a rectangle are right angles. So, SOLUTION:

All four angles of a rectangle are right angles. So,

ANSWER: 50 The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. 27. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 The measures of angles 2 and 3 are equal as they and 6 is an isosceles triangle. Therefore, m∠6 = are alternate interior angles. m∠4 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 ANSWER: and 8 is an isosceles triangle. So, m∠7 = m∠3. 50 Therefore, m 7 = m 2 = 40. ∠ ∠ 31. ANSWER: SOLUTION: 40 The measures of angles 2 and 3 are equal as they are alternate interior angles. 28. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 The measures of angles 2 and 3 are equal as they and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. are alternate interior angles. The sum of the three angles of a triangle is 180. Therefore, m∠3 = m∠2 = 40. Therefore, m∠8 = 180 – (40 + 40) = 100. ANSWER: ANSWER: 40 100

29. 32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one SOLUTION: shown. To what length should she cut the metal All four angles of a rectangle are right angles. So, supports in order for the bookshelf to be square, which means that the angles formed by the shelves and the vertical supports are all right angles? Explain your reasoning. The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. SOLUTION: ANSWER: In order for the bookshelf to be square, the lengths of 80 all of the metal supports should be equal. Since we know the length of the bookshelves and the distance 30. between them, use the Pythagorean Theorem to find SOLUTION: the lengths of the metal supports. All four angles of a rectangle are right angles. So, Let x be the length of each support. Express 3 feet as 36 inches.

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m 4 = m 1 = 50. ∠ ∠ Therefore, the metal supports should each be 39 Since the diagonals of a rectangle are congruent and inches long or 3 feet 3 inches long. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = ANSWER: m∠4 = 50. 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports ANSWER: should be equal. Since I knew the length of the 50 bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the 31. metal supports. SOLUTION: PROOF Write a two-column proof. The measures of angles 2 and 3 are equal as they 33. Theorem 6.13 are alternate interior angles. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 You need to walk through the proof step by step. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. Look over what you are given and what you need to The sum of the three angles of a triangle is 180. prove. Here, you are given WXYZ is a rectangle with Therefore, m∠8 = 180 – (40 + 40) = 100. diagonals . You need to prove 6-4 RectanglesANSWER: . Use the properties that you have learned 100 about rectangles to walk through the proof.

Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of SOLUTION: rectangle) In order for the bookshelf to be square, the lengths of 5. (All right .) all of the metal supports should be equal. Since we 6. (SAS) know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find 7. (CPCTC) the lengths of the metal supports. Let x be the length of each support. Express 3 feet ANSWER: as 36 inches. Given: WXYZ is a rectangle with diagonals . Prove:

Therefore, the metal supports should each be 39

inches long or 3 feet 3 inches long. Proof: ANSWER: 3 ft 3 in.; Sample answer: In order for the bookshelf 1. WXYZ is a rectangle with diagonals . to be square, the lengths of all of the metal supports (Given) should be equal. Since I knew the length of the 2. (Opp. sides of a .) bookshelves and the distance between them, I used 3. (Refl. Prop.) the Pythagorean Theorem to find the lengths of the metal supports. 4. are right angles. (Def. of rectangle) PROOF Write a two-column proof. 5. (All right .) 33. Theorem 6.13 6. (SAS) SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. Look over what you are given and what you need to 34. Theorem 6.14 prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove SOLUTION: . Use the properties that you have learned You need to walk through the proof step by step. Look over what you are given and what you need to about rectangles to walk through the proof. prove. Here, you are given

Given: WXYZ is a rectangle with diagonals . You need to eSolutions Manual - Powered by Cognero prove that WXYZ is a rectangle.. Use the propertiesPage 11 . that you have learned about rectangles to walk Prove: through the proof.

Given: Prove: WXYZ is a rectangle.

Proof: 1. WXYZ is a rectangle with diagonals . (Given)

. 8. are right angles. (If

Prove: 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

ANSWER: Proof: Given: 1. WXYZ is a rectangle with diagonals . Prove: WXYZ is a rectangle. (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) 5. (All right .) Proof: 6. (SAS) 1. (Given) 7. (CPCTC) 2. (SSS) 3. (CPCTC) 34. Theorem 6.14 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. SOLUTION: sides are , then quad. is .) You need to walk through the proof step by step. 6. are supplementary. (Cons. Look over what you are given and what you need to are suppl.) prove. Here, you are given 7. (Def of suppl.) . You need to 8. are right angles. (If prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk 2 and suppl., each is a rt. .) through the proof. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) Given: 10. WXYZ is a rectangle. (Def. of rectangle) Prove: WXYZ is a rectangle. PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION:

9. are right angles. (If a has 1 ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right rt. , it has 4 rt. .) angle, then it has four right angles. So by the 10. WXYZ is a rectangle. (Def. of rectangle) definition of a rectangle, ABCD is a rectangle.

ANSWER: ANSWER: Given: Prove: WXYZ is a rectangle.

ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right Proof: angle, then it has four right angles. So by the

1. (Given) definition of a rectangle, ABCD is a rectangle. 2. (SSS) 3. (CPCTC) 36. If a quadrilateral has four right angles, then it is a 4. (Def. of ) rectangle. 5. WXYZ is a parallelogram. (If both pairs of opp. SOLUTION: sides are , then quad. is .) You need to walk through the proof step by step. 6. are supplementary. (Cons. Look over what you are given and what you need to are suppl.) prove. Here, you are given a quadrilateral with four 7. (Def of suppl.) right angles. You need to prove that it is a rectangle. 8. are right angles. (If Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and

2 and suppl., each is a rt. .) rectangles to walk through the proof. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. ABCD is a quadrilateral with four right angles. SOLUTION: ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a You need to walk through the proof step by step. rectangle, ABCD is a rectangle. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one ANSWER: right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

Step 2: Place the compass on point P. Using the ABCD is a parallelogram, and is a right angle. same compass setting, draw an arc to the left and Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the right of P that intersects . definition of a rectangle, ABCD is a rectangle. Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. 36. If a quadrilateral has four right angles, then it is a Place the compass on each point of intersection and rectangle. draw arcs that intersect above . SOLUTION: You need to walk through the proof step by step. Step 4: Draw line a through point P and the Look over what you are given and what you need to intersection of the arcs drawn in step 3. prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Step 5: Repeat steps 2 and 3 using point Q. Let ABCD be a quadrilateral with four right angles.

Then use the properties of parallelograms and rectangles to walk through the proof. Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Since and , the measure of 37. CONSTRUCTION Construct a rectangle using the angles P and Q is 90 degrees. Lines that are construction for congruent segments and the perpendicular to the same line are parallel, so construction for a line perpendicular to another line through a point on the line. Justify each step of the . The same compass setting was used to construction. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and SOLUTION: congruent, then the quadrilateral is a parallelogram. Sample answer: A parallelogram with right angles is a rectangle. Step 1: Graph a line and place points P and Q on the Thus, PRSQ is a rectangle. line.

Step 2: Place the compass on point P. Using the ANSWER: same compass setting, draw an arc to the left and right of P that intersects . Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that Step 3: Open the compass to a setting greater than are perpendicular to the same line are parallel, so the distance from P to either point of intersection. Place the compass on each point of intersection and . The same compass setting was used to draw arcs that intersect above . locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 4: Draw line a through point P and the A parallelogram with right angles is a rectangle.

intersection of the arcs drawn in step 3. Thus, PRSQ is a rectangle.

Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S. 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Step 9: Draw . Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then the end zone is a rectangle. Since and , the measure of ANSWER: angles P and Q is 90 degrees. Lines that are Sample answer: He should measure the diagonals of perpendicular to the same line are parallel, so the end zone and both pairs of opposite sides. If the . The same compass setting was used to diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and ALGEBRA Quadrilateral WXYZ is a rectangle. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

ANSWER: 39. If XW = 3, WZ = 4, and XZ = b, find YW.

SOLUTION: Sample answer: Since and , the The diagonals of a rectangle are congruent to each measure of angles P and Q is 90 degrees. Lines that other. So, YW = XZ = b. All four angles of a are perpendicular to the same line are parallel, so rectangle are right angles. So, is a right . The same compass setting was used to triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 locate points R and S, so . If one pair of WZ . opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Therefore, YW = 5. ANSWER: 5

wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Therefore, WY = 2(5) = 10. Sample answer: If the diagonals of a parallelogram ANSWER: are congruent, then it is a rectangle. He should 10 measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides 41. SIGNS The sign below is in the foyer of Nyoko’s are congruent, then the quadrilateral is a school. Based on the dimensions given, can Nyoko parallelogram. If the diagonals are congruent, then be sure that the sign is a rectangle? Explain your the end zone is a rectangle. reasoning. ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

SOLUTION:

Both pairs of opposite sides are congruent, so the

Therefore, YW = 5. PROOF Write a coordinate proof of each statement. ANSWER: 42. The diagonals of a rectangle are congruent. 5 SOLUTION: Begin by positioning quadrilateral ABCD on a 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. coordinate plane. Place vertex D at the origin. Let SOLUTION: the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, The diagonals of a rectangle are congruent to each b), and C(a, 0). You need to walk through the proof other. So, WY = XZ = 2c. All four angles of a step by step. Look over what you are given and what rectangle are right angles. So, is a right you need to prove. Here, you are given ABCD is a 2 2 triangle. By the Pythagorean Theorem, XZ = XY + rectangle and you need to prove . Use the 2 ZY . properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove:

Therefore, WY = 2(5) = 10.

ANSWER: 10

The diagonals, have the same length, so they are congruent.

ANSWER:

No; sample answer: Both pairs of opposite sides are Proof: congruent, so the sign is a parallelogram, but no Use the Distance Formula to find measure is given that can be used to prove that it is a rectangle. and . have the same length, so they are PROOF Write a coordinate proof of each congruent. statement. 42. The diagonals of a rectangle are congruent. 43. If the diagonals of a parallelogram are congruent, SOLUTION: then it is a rectangle. Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let SOLUTION: the length of the bases be a units and the height be b Begin by positioning parallelogram ABCD on a units. Then the rest of the vertices are A(0, b), B(a, coordinate plane. Place vertex A at the origin. Let b), and C(a, 0). You need to walk through the proof the length of the bases be a units and the height be c step by step. Look over what you are given and what units. Then the rest of the vertices are B(a, 0), C(b + you need to prove. Here, you are given ABCD is a a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given rectangle and you need to prove . Use the and what you need to prove. Here, you are given properties that you have learned about rectangles to walk through the proof. ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you Given: ABCD is a rectangle. have learned about rectangles to walk through the

Prove: proof.

Given: ABCD and Prove: ABCD is a rectangle.

Proof: Use the Distance Formula to find the lengths of the diagonals. Proof:

The diagonals, have the same length, so AC = BD. So, they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.

Proof: ANSWER: Use the Distance Formula to find Given: ABCD and and . Prove: ABCD is a rectangle. have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a Proof: coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given But AC = BD. So, and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

AC = BD. So, label the intersections R. b. TABULAR Use a protractor to measure the

appropriate angles and complete the table below.

ANSWER: Given: ABCD and Prove: ABCD is a rectangle.

b. Use a protractor to measure each angle listed in the table. Proof:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each

45. CHALLENGE In rectangle ABCD, Find the c. VERBAL Make a conjecture about the diagonals values of x and y. of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and b. Use a protractor to measure each angle listed in bisect each other. So, is an isosceles triangle the table. with So,

ANSWER:

x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. b. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a c. Sample answer: The diagonals of a parallelogram single vertex of a triangle. with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

SOLUTION:

All four angles of a rectangle are right angles. So, So, Tamika is correct. ANSWER: Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of The diagonals of a rectangle are congruent and the angles are formed by a single vertex of a triangle. bisect each other. So, is an isosceles triangle In order for the quadrilateral to be a rectangle, one of with the angles in the congruent triangles has to be a right So, angle.

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines. ANSWER: 6 The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the 48. OPEN ENDED Write the equations of four lines slope of DC is 0. Since a pair of sides of the having intersections that form the vertices of a quadrilateral is both parallel and congruent, by rectangle. Verify your answer using coordinate Theorem 6.12, the quadrilateral is a parallelogram. geometry. Since is horizontal and is vertical, the lines SOLUTION: are perpendicular and the measure of the angle they Sample answer: form is 90. By Theorem 6.6, if a parallelogram has A rectangle has 4 right angles and each pair of one right angle, it has four right angles. Therefore, by opposite sides is parallel and congruent. Lines that definition, the parallelogram is a rectangle. run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. 49. WRITING IN MATH Explain why all rectangles Selecting y = 0 and x = 0 will form 2 sides of the are parallelograms, but all parallelograms are not rectangle. The lines y = 4 and x = 6 form the other 2 rectangles.

sides. SOLUTION:

Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

ANSWER:

Parallelograms with right angles are rectangles, so 51. ALGEBRA A rectangular playground is surrounded some parallelograms are rectangles, but others with by an 80-foot fence. One side of the playground is 10 non-right angles are not. feet longer than the other. Which of the following equations could be used to find r, the shorter side of ANSWER: the playground? Sample answer: All rectangles are parallelograms F 10r + r = 80 because, by definition, both pairs of opposite sides G 4r + 10 = 80 are parallel. Parallelograms with right angles are H rectangles, so some parallelograms are rectangles, r(r + 10) = 80 but others with non-right angles are not. J 2(r + 10) + 2r = 80 SOLUTION: 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram The formula to find the perimeter of a rectangle is FGHJ a rectangle? given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. Therefore, the correct choice is J.

ANSWER:

J A x = 3, y = 4 B x = 4, y = 3 52. SHORT RESPONSE What is the measure of C x = 7, y = 8 ? D x = 8, y = 7 SOLUTION: The opposite sides of a parallelogram are congruent. So, FJ = GH. –3x + 5y = 11 FJ = GH

By the Exterior Angle Theorem, Add the two equations to eliminate the x-term and solve for y. –3x + 5y = 11 ANSWER: 3x + y = 13 112 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6. 53. SAT/ACT If p is odd, which of the following must also be odd?

Use the value of y to find the value of x. A 2p 3x + 4 = 13 Substitute. B 2p + 2 3x = 9 Subtract 4 from each side.

x = 3 Divide each side by 3. C

SOLUTION: Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a The formula to find the perimeter of a rectangle is given by P = 2(l + w). The perimeter is given to be 8 multiple of 2 so it is even. ft. The width is r and the length is r + 10. So, 2r + 2 Choice E p + 2 (r + 10) = 80. Adding 2 to an odd number results in an odd Therefore, the correct choice is J. number. This is odd. The correct choice is E. ANSWER: J ANSWER: E 52. SHORT RESPONSE What is the measure of ? ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

ANSWER: 112

53. SAT/ACT If p is odd, which of the following must also be odd? A 2p ANSWER: B 2p + 2 x = 2, y = 41 C D E p + 2 SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. 55. Choice A 2p SOLUTION: This is a multiple of 2 so it is even. The opposite sides of a parallelogram are congruent. Choice B 2p + 2

This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

55. Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5). SOLUTION: The opposite sides of a parallelogram are congruent. ANSWER: (2.5, 0.5) Refer to the figure. The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

58. If , name two congruent angles. ANSWER: x = 8, y = 22 SOLUTION: Since is an isosceles triangle. So,

ANSWER:

56. SOLUTION: 59. If , name two congruent segments. The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. SOLUTION: Solve the first equation to find the value of y. In a triangle with two congruent angles, the sides 2y = 14 opposite to the congruent angles are also congruent. y = 7 So, . Use the value of y in the second equation and solve for x. ANSWER: 7 + 3 = 2x + 6 4 = 2x 2 = x 60. If , name two congruent segments. ANSWER: SOLUTION: x = 2, y = 7 In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. 57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of So, . parallelogram ABCD with vertices A(1, 3), B(6, 2), C ANSWER: (4, –2), and D(–1, –1).

SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem, FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. Therefore, If feet, PS = 7 feet, and ANSWER: , find each measure. 33.5

4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore,

1. QR ANSWER: SOLUTION: 56.5 The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft. ALGEBRA Quadrilateral DEFG is a rectangle.

ANSWER: 7 ft

2. SQ 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle bisect each other. So, SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: ANSWER: 11

6. If ,

3. find . SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, The diagonals of a rectangle are congruent and

bisect each other. So, is an isosceles triangle.

Then, By the Exterior Angle Theorem,

Therefore,

ANSWER: ANSWER: 33.5 51

4. 7. PROOF If ABDE is a rectangle and ,

SOLUTION: prove that . The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore,

Given: ABDE is a rectangle; 5. If FD = 3x – 7 and EG = x + 5, find EG. Prove: SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. Statements(Reasons)

EG = 6 + 5 = 11 1. ABDE is a rectangle; . ANSWER: (Given) 2. ABDE is a parallelogram. (Def. of 11 rectangle) 3. (Opp. sides of a .) 6. If , 4. are right angles.(Def. of rectangle) find . 5. (All rt .) SOLUTION: 6. (SAS) All four angles of a rectangle are right angles. So, 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

ANSWER: 51

SOLUTION: Given: ABDE is a rectangle; Prove:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) Use the slope formula to find the slope of the sides of 2. ABDE is a parallelogram. (Def. of the quadrilateral. rectangle)

3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; There are no parallel sides, so WXYZ is not a Prove: parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION: SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral. First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

11. DB so ABCD is a SOLUTION: parallelogram. All four angles of a rectangle are right angles. So, A parallelogram is a rectangle if the diagonals are is a right triangle. By the Pythagorean congruent. Use the Distance formula to find the 2 2 2 lengths of the diagonals. Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

So the diagonals are congruent. Thus, ABCD is a rectangle.

By the Vertical Angle Theorem,

10. BC SOLUTION: ANSWER: The opposite sides of a rectangle are parallel and 25 congruent. Therefore, BC = AD = 2 ft. CCSS REGULARITY Quadrilateral WXYZ is a ANSWER: rectangle. 2 ft

11. DB SOLUTION: All four angles of a rectangle are right angles. So, 14. If ZY = 2x + 3 and WX = x + 4, find WX. is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . SOLUTION: BC = AD = 2 The opposite sides of a rectangle are parallel and DC = AB = 6 congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5. ANSWER: ANSWER: 6.3 ft 5 12. 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, is an isosceles The diagonals of a rectangle bisect each other. So, triangle. PY = WP. Then,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. By the Vertical Angle Theorem,

ANSWER: 50 ANSWER: 13. 43 SOLUTION: 16. If , find The diagonals of a rectangle are congruent and . bisect each other. So, is an isosceles SOLUTION: triangle. All four angles of a rectangle are right angles. So, Then, All four angles of a rectangle are right angles.

The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 48

19. If , find . SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 43 is a right triangle. Then, 16. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: ANSWER: 48 Proof: Statements (Reasons) 19. If , find 1. ABCD is a rectangle. (Given) . 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) SOLUTION:

All four angles of a rectangle are right angles. So, 4. (Refl. Prop.) is a right triangle. Then, 5. (Diagonals of a rectangle are .) 6. (SSS)

21. Given: QTVW is a rectangle.

Given: QTVW is a rectangle.

Statements (Reasons) 3. (Opp sides of a .) 1. ABCD is a rectangle. (Given) 4. are right angles.(Def. of rectangle) 2. ABCD is a parallelogram. (Def. of rectangle) 5. (All rt .) 3. (Opp. sides of a .) 6. QR = ST (Def. of segs.) 4. (Refl. Prop.) 7. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 8. RS = RS (Def. of segs.) 6. (SSS) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) ANSWER: 11. QS = RT (Subst.) Proof: 12. (Def. of segs.) Statements (Reasons)

1. ABCD is a rectangle. (Given) 13. (SAS) 2. ABCD is a parallelogram. (Def. of rectangle) ANSWER:

Given: QTVW is a rectangle. SOLUTION:

Prove: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle)

Statements (Reasons) –1. So, any two adjacent sides are perpendicular to 1. QTVW is a rectangle; .(Given) each other. That is, all the four angles are right 2. QTVW is a parallelogram. (Def. of rectangle) angles. Therefore, WXYZ is a rectangle. 3. (Opp sides of a .) ANSWER: 4. are right angles.(Def. of rectangle) 5. (All rt .) Yes; slope of slope of , slope 6. QR = ST (Def. of segs.) of slope of . So WXYZ is a 7. (Refl. Prop.) parallelogram. The product of the slopes of 8. RS = RS (Def. of segs.) consecutive sides is –1, so the consecutive sides are 9. QR + RS = RS + ST (Add. prop.) perpendicular and form right angles. Thus, WXYZ is a 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) rectangle. 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

ANSWER:

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION: SOLUTION:

First, Use the Distance formula to find the lengths of First, use the Distance Formula to find the lengths of the sides of the quadrilateral. the sides of the quadrilateral.

So the diagonals are not congruent. Thus, JKLM is So the diagonals are congruent. Thus, QRST is a not a rectangle. rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION: 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, use the slope formula to find the lengths of the First, use the Distance Formula to find the lengths of sides of the quadrilateral. the sides of the quadrilateral.

So the diagonals are congruent. Thus, QRST is a parallelogram. rectangle. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. ANSWER: Therefore, GHJK is not a rectangle. Yes; so QRST is a ANSWER: parallelogram. so the diagonals are No; slope of slope of and slope of congruent. QRST is a rectangle. slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

26. SOLUTION: First, use the slope formula to find the lengths of the sides of the quadrilateral. All four angles of a rectangle are right angles. So,

ANSWER: 50

27. SOLUTION: The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. The measures of angles 2 and 3 are equal as they

Therefore, the quadrilateral GHJK is a are alternate interior angles. parallelogram. Since the diagonals of a rectangle are congruent and None of the adjacent sides have slopes whose bisect each other, the triangle with the angles 3, 7 product is –1. So, the angles are not right angles. and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, GHJK is not a rectangle. Therefore, m∠7 = m∠2 = 40.

29. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and Therefore, the metal supports should each be 39 bisect each other, the triangle with the angles 4, 5 inches long or 3 feet 3 inches long. and 6 is an isosceles triangle. Therefore, m 6 = ∠ ANSWER: m∠4 = 50. 3 ft 3 in.; Sample answer: In order for the bookshelf ANSWER: to be square, the lengths of all of the metal supports 50 should be equal. Since I knew the length of the bookshelves and the distance between them, I used 31. the Pythagorean Theorem to find the lengths of the metal supports. SOLUTION: The measures of angles 2 and 3 are equal as they PROOF Write a two-column proof. are alternate interior angles. 33. Theorem 6.13 Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 You need to walk through the proof step by step. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. Look over what you are given and what you need to The sum of the three angles of a triangle is 180. prove. Here, you are given WXYZ is a rectangle with Therefore, m∠8 = 180 – (40 + 40) = 100. diagonals . You need to prove ANSWER: . Use the properties that you have learned 100 about rectangles to walk through the proof.

32. CCSS MODELING Jody is building a new Given: WXYZ is a rectangle with diagonals bookshelf using wood and metal supports like the one . shown. To what length should she cut the metal supports in order for the bookshelf to be square, Prove: which means that the angles formed by the shelves and the vertical supports are all right angles? Explain your reasoning.

Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .)

Therefore, the metal supports should each be 39 inches long or 3 feet 3 inches long.

ANSWER: Proof: 3 ft 3 in.; Sample answer: In order for the bookshelf 1. WXYZ is a rectangle with diagonals . to be square, the lengths of all of the metal supports (Given) should be equal. Since I knew the length of the 2. (Opp. sides of a .) bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the 3. (Refl. Prop.) metal supports. 4. are right angles. (Def. of rectangle) PROOF Write a two-column proof. 5. (All right .) 33. Theorem 6.13 6. (SAS) SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with 34. Theorem 6.14 diagonals . You need to prove SOLUTION: . Use the properties that you have learned You need to walk through the proof step by step. about rectangles to walk through the proof. Look over what you are given and what you need to prove. Here, you are given Given: WXYZ is a rectangle with diagonals . You need to . prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk Prove: through the proof.

Given: Prove: WXYZ is a rectangle.

7. (CPCTC) sides are , then quad. is .) 6. are supplementary. (Cons. ANSWER:

Given: WXYZ is a rectangle with diagonals are suppl.) 7. (Def of suppl.) . 8. are right angles. (If Prove: 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

Proof: ANSWER:

1. WXYZ is a rectangle with diagonals . Given: (Given) Prove: WXYZ is a rectangle. 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of

rectangle) 5. (All right .) Proof: 6. (SAS) 1. (Given) 7. (CPCTC) 2. (SSS) 3. (CPCTC) 34. Theorem 6.14 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. SOLUTION: sides are , then quad. is .) 6-4 RectanglesYou need to walk through the proof step by step. Look over what you are given and what you need to 6. are supplementary. (Cons. prove. Here, you are given are suppl.) . You need to 7. (Def of suppl.) prove that WXYZ is a rectangle.. Use the properties 8. are right angles. (If that you have learned about rectangles to walk 2 and suppl., each is a rt. .) through the proof. 9. are right angles. (If a has 1

3. (CPCTC) Let ABCD be a parallelogram with one right angle. 4. (Def. of ) Then use the properties of parallelograms to walk 5. WXYZ is a parallelogram. (If both pairs of opp. through the proof. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 ABCD is a parallelogram, and is a right angle. rt. , it has 4 rt. .) Since ABCD is a parallelogram and has one right 10. WXYZ is a rectangle. (Def. of rectangle) angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. ANSWER: ANSWER: Given: Prove: WXYZ is a rectangle.

ABCD is a parallelogram, and is a right angle. Proof: Since ABCD is a parallelogram and has one right 1. (Given) angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. 2. (SSS) 3. (CPCTC) 36. If a quadrilateral has four right angles, then it is a 4. (Def. of ) rectangle. 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) SOLUTION: 6. are supplementary. (Cons. You need to walk through the proof step by step. Look over what you are given and what you need to are suppl.) prove. Here, you are given a quadrilateral with four 7. (Def of suppl.) right angles. You need to prove that it is a rectangle. eSolutions8. Manual - Powered by Cognero are right angles. (If Let ABCD be a quadrilateral with four right angles.Page 12 2 and suppl., each is a rt. .) Then use the properties of parallelograms and rectangles to walk through the proof. 9. are right angles. (If a has 1

rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

You need to walk through the proof step by step. Look over what you are given and what you need to Step 4: Draw line a through point P and the prove. Here, you are given a quadrilateral with four intersection of the arcs drawn in step 3. right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. Step 5: Repeat steps 2 and 3 using point Q. Then use the properties of parallelograms and rectangles to walk through the proof. Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Since and , the measure of 37. CONSTRUCTION Construct a rectangle using the angles P and Q is 90 degrees. Lines that are construction for congruent segments and the perpendicular to the same line are parallel, so construction for a line perpendicular to another line through a point on the line. Justify each step of the . The same compass setting was used to construction. locate points R and S, so . If one pair of SOLUTION: opposite sides of a quadrilateral is both parallel and Sample answer: congruent, then the quadrilateral is a parallelogram. Step 1: Graph a line and place points P and Q on the A parallelogram with right angles is a rectangle. line. Thus, PRSQ is a rectangle.

Step 2: Place the compass on point P. Using the ANSWER: same compass setting, draw an arc to the left and

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that

intersects line b. Label the point of intersection S. 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for Step 9: Draw . painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then the end zone is a rectangle. Since and , the measure of ANSWER: angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the . The same compass setting was used to diagonals and both pairs of opposite sides are locate points R and S, so . If one pair of congruent, then the end zone is a rectangle. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. ALGEBRA Quadrilateral WXYZ is a rectangle. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

ANSWER: 39. If XW = 3, WZ = 4, and XZ = b, find YW.

SOLUTION: Sample answer: Since and , the The diagonals of a rectangle are congruent to each measure of angles P and Q is 90 degrees. Lines that other. So, YW = XZ = b. All four angles of a are perpendicular to the same line are parallel, so rectangle are right angles. So, is a right . The same compass setting was used to triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 locate points R and S, so . If one pair of WZ . opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Therefore, YW = 5.

ANSWER: 5

ALGEBRA Quadrilateral WXYZ is a rectangle.

SOLUTION:

Therefore, YW = 5. PROOF Write a coordinate proof of each statement. ANSWER: 42. The diagonals of a rectangle are congruent. 5 SOLUTION: 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let SOLUTION: the length of the bases be a units and the height be b The diagonals of a rectangle are congruent to each units. Then the rest of the vertices are A(0, b), B(a, other. So, WY = XZ = 2c. All four angles of a b), and C(a, 0). You need to walk through the proof rectangle are right angles. So, is a right step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a triangle. By the Pythagorean Theorem, XZ2 = XY2 + 2 rectangle and you need to prove . Use the ZY . properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove:

Therefore, WY = 2(5) = 10.

ANSWER: 10

The diagonals, have the same length, so they are congruent.

ANSWER: SOLUTION: Given: ABCD is a rectangle.

PROOF Write a coordinate proof of each have the same length, so they are statement. congruent. 42. The diagonals of a rectangle are congruent. 43. If the diagonals of a parallelogram are congruent, SOLUTION: then it is a rectangle. Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let SOLUTION: the length of the bases be a units and the height be b Begin by positioning parallelogram ABCD on a units. Then the rest of the vertices are A(0, b), B(a, coordinate plane. Place vertex A at the origin. Let b), and C(a, 0). You need to walk through the proof the length of the bases be a units and the height be c step by step. Look over what you are given and what units. Then the rest of the vertices are B(a, 0), C(b + you need to prove. Here, you are given ABCD is a a, c), and D(b, c). You need to walk through the rectangle and you need to prove . Use the proof step by step. Look over what you are given properties that you have learned about rectangles to and what you need to prove. Here, you are given walk through the proof. ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you Given: ABCD is a rectangle. have learned about rectangles to walk through the Prove: proof.

Given: ABCD and Prove: ABCD is a rectangle.

Proof: Use the Distance Formula to find the lengths of the diagonals. Proof:

The diagonals, have the same length, so AC = BD. So,

they are congruent. ANSWER: Given: ABCD is a rectangle. Prove:

and . Given: ABCD and have the same length, so they are Prove: ABCD is a rectangle. congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle.

SOLUTION: Begin by positioning parallelogram ABCD on a Proof: coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given But AC = BD. So, and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

ANSWER:

Given: ABCD and Prove: ABCD is a rectangle.

b. Use a protractor to measure each angle listed in Proof: the table.

c. Sample answer: Each of the angles listed in the

But AC = BD. So, table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: a. Sample answer:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined

and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

CHALLENGE 45. In rectangle ABCD, c. VERBAL Make a conjecture about the diagonals Find the of a parallelogram with four congruent sides. values of x and y. SOLUTION: a. Sample answer:

SOLUTION: All four angles of a rectangle are right angles. So,

b. Use a protractor to measure each angle listed in The diagonals of a rectangle are congruent and the table. bisect each other. So, is an isosceles triangle with So,

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. b. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a c. Sample answer: The diagonals of a parallelogram single vertex of a triangle. with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

SOLUTION: All four angles of a rectangle are right angles. So, So, Tamika is correct.

ANSWER:

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: So, Tamika is correct. A rectangle has 4 right angles and each pair of ANSWER: opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other Tamika; Sample answer: When two congruent and perpendicular to lines parallel to the y-axis. triangles are arranged to form a quadrilateral, two of Selecting y = 0 and x = 0 will form 2 sides of the the angles are formed by a single vertex of a triangle. rectangle. The lines y = 4 and x = 6 form the other 2 In order for the quadrilateral to be a rectangle, one of sides. the angles in the congruent triangles has to be a right angle. x = 0, x = 6, y = 0, y = 4; 47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines?

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the SOLUTION: quadrilateral is both parallel and congruent, by Let the lines n, p, q, and r intersect the line l at the Theorem 6.12, the quadrilateral is a parallelogram. points A, B, C, and D respectively. Similarly, let the Since is horizontal and is vertical, the lines lines intersect the line m at the points W, X, Y, and Z. are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

Theorem 6.12, the quadrilateral is a parallelogram. but others with non-right angles are not. Since is horizontal and is vertical, the lines 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = are perpendicular and the measure of the angle they 13, what values of x and y make parallelogram form is 90. By Theorem 6.6, if a parallelogram has FGHJ a rectangle? one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

If the diagonals of a parallelogram are congruent, The length of is 6 − 0 or 6 units and the length of then it is a rectangle. Since FGHJ is a parallelogram, is 6 – 0 or 6 units. The slope of AB is 0 and the the diagonals bisect each other. So, if it is a slope of DC is 0. Since a pair of sides of the rectangle, then will be an isosceles triangle quadrilateral is both parallel and congruent, by with That is, 3x + y = 13. Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines Add the two equations to eliminate the x-term and are perpendicular and the measure of the angle they solve for y. form is 90. By Theorem 6.6, if a parallelogram has –3x + 5y = 11 one right angle, it has four right angles. Therefore, by 3x + y = 13 definition, the parallelogram is a rectangle. 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6. 49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not Use the value of y to find the value of x. rectangles. 3x + 4 = 13 Substitute. SOLUTION: 3x = 9 Subtract 4 from each side. Sample answer: A parallelogram is a quadrilateral in x = 3 Divide each side by 3. which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs So, x = 3, y = 4, the correct choice is A. of opposite sides parallel. So rectangles are always ANSWER: parallelograms. A Parallelograms with right angles are rectangles, so 51. ALGEBRA A rectangular playground is surrounded some parallelograms are rectangles, but others with by an 80-foot fence. One side of the playground is 10 non-right angles are not. feet longer than the other. Which of the following ANSWER: equations could be used to find r, the shorter side of

Sample answer: All rectangles are parallelograms the playground? because, by definition, both pairs of opposite sides F 10r + r = 80 are parallel. Parallelograms with right angles are G 4r + 10 = 80 rectangles, so some parallelograms are rectangles, H r(r + 10) = 80 but others with non-right angles are not. J 2(r + 10) + 2r = 80 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = SOLUTION: 13, what values of x and y make parallelogram The formula to find the perimeter of a rectangle is FGHJ a rectangle? given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. Therefore, the correct choice is J.

ANSWER: A x = 3, y = 4 J B x = 4, y = 3 SHORT RESPONSE C x = 7, y = 8 52. What is the measure of

D x = 8, y = 7 ? SOLUTION: The opposite sides of a parallelogram are congruent. So, FJ = GH. –3x + 5y = 11 FJ = GH

x = 3 Divide each side by 3. C

So, x = 3, y = 4, the correct choice is A. D E p + 2 ANSWER: A SOLUTION: Analyze each answer choice to determine which 51. ALGEBRA A rectangular playground is surrounded expression is odd when p is odd. by an 80-foot fence. One side of the playground is 10 Choice A 2p feet longer than the other. Which of the following This is a multiple of 2 so it is even. equations could be used to find r, the shorter side of Choice B 2p + 2 the playground? This can be rewritten as 2(p + 1) which is also a F 10r + r = 80 multiple of 2 so it is even. G 4r + 10 = 80 H r(r + 10) = 80 Choice C J 2(r + 10) + 2r = 80 If p is odd , is not a whole number. SOLUTION: Choice D 2p – 2 The formula to find the perimeter of a rectangle is This can be rewritten as 2(p - 1) which is also a given by P = 2(l + w). The perimeter is given to be 8 multiple of 2 so it is even. ft. The width is r and the length is r + 10. So, 2r + 2 Choice E p + 2 (r + 10) = 80. Adding 2 to an odd number results in an odd Therefore, the correct choice is J. number. This is odd. The correct choice is E. ANSWER: J ANSWER: E 52. SHORT RESPONSE What is the measure of ? ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

54. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and The opposite sides of a parallelogram are congruent. bisect each other. So, is an isosceles triangle with

So, By the Exterior Angle Theorem, In a parallelogram, consecutive interior angles are

supplementary. ANSWER: 112

multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. The correct choice is E. ANSWER: x = 8, y = 22 ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram. 56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. 54. Solve the first equation to find the value of y. SOLUTION: 2y = 14 The opposite sides of a parallelogram are congruent. y = 7 Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x In a parallelogram, consecutive interior angles are 2 = x supplementary. ANSWER: x = 2, y = 7

55. Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5). SOLUTION: The opposite sides of a parallelogram are congruent. ANSWER: (2.5, 0.5)

ANSWER: 58. If , name two congruent angles. x = 8, y = 22 SOLUTION: Since is an isosceles triangle. So,

ANSWER: 56. SOLUTION: The diagonals of a parallelogram bisect each other. 59. If , name two congruent segments. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. SOLUTION:

Solve the first equation to find the value of y. In a triangle with two congruent angles, the sides

2y = 14 opposite to the congruent angles are also congruent. y = 7 Use the value of y in the second equation and solve So, . for x. ANSWER: 7 + 3 = 2x + 6

4 = 2x 2 = x 60. If , name two congruent segments. ANSWER: SOLUTION: x = 2, y = 7 In a triangle with two congruent angles, the sides 57. COORDINATE GEOMETRY Find the opposite to the congruent angles are also congruent. coordinates of the intersection of the diagonals of So, . parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). ANSWER:

SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

FARMING An X-brace on a rectangular barn

door is both decorative and functional. It helps Therefore, to prevent the door from warping over time. If feet, PS = 7 feet, and ANSWER: 33.5 , find each measure. 4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore,

1. QR ANSWER: 56.5 SOLUTION: The opposite sides of a rectangle are parallel and ALGEBRA Quadrilateral DEFG is a rectangle. congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: SOLUTION: The diagonals of a rectangle bisect each other. So, The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 11 ANSWER:

6. If , find .

3. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Given: ABDE is a rectangle; Prove: 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

Statements(Reasons) Use the value of x to find EG.

EG = 6 + 5 = 11 1. ABDE is a rectangle; . (Given) ANSWER: 2. ABDE is a parallelogram. (Def. of 11 rectangle) 3. (Opp. sides of a .) 6. If , 4. are right angles.(Def. of rectangle) find . 5. (All rt .) 6. (SAS) SOLUTION:

All four angles of a rectangle are right angles. So, 7. (CPCTC) ANSWER:

Given: ABDE is a rectangle; Prove:

ANSWER: 51

7. PROOF If ABDE is a rectangle and , Statements(Reasons) prove that . 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. Look over what you are given and what you need to COORDINATE GEOMETRY Graph each prove. You are given ABDE is a rectangle and quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your . You need to prove . Use the answer using the indicated formula. properties that you have learned about rectangles to 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula walk through the proof. SOLUTION: Given: ABDE is a rectangle; Prove:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) Use the slope formula to find the slope of the sides of 2. ABDE is a parallelogram. (Def. of the quadrilateral. rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; There are no parallel sides, so WXYZ is not a Prove: parallelogram. Therefore, WXYZ is not a rectangle. ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of

Use the slope formula to find the slope of the sides of First, Use the Distance formula to find the lengths of the quadrilateral. the sides of the quadrilateral.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals. There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

11. DB SOLUTION: so ABCD is a parallelogram. All four angles of a rectangle are right angles. So, A parallelogram is a rectangle if the diagonals are is a right triangle. By the Pythagorean 2 2 2 congruent. Use the Distance formula to find the Theorem, DB = BC + DC . lengths of the diagonals. BC = AD = 2 DC = AB = 6

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a 6.3 ft parallelogram. , so the diagonals are 12. congruent. Thus, ABCD is a rectangle. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

By the Vertical Angle Theorem,

10. BC ANSWER: SOLUTION: 25 The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. CCSS REGULARITY Quadrilateral WXYZ is a rectangle. ANSWER: 2 ft

11. DB SOLUTION: All four angles of a rectangle are right angles. So, 14. If ZY = 2x + 3 and WX = x + 4, find WX. is a right triangle. By the Pythagorean SOLUTION: 2 2 2 Theorem, DB = BC + DC . The opposite sides of a rectangle are parallel and BC = AD = 2 congruent. So, ZY = WX. DC = AB = 6 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5.

ANSWER: ANSWER: 6.3 ft 5

12. 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and The diagonals of a rectangle bisect each other. So, bisect each other. So, is an isosceles PY = WP. triangle. Then, The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

By the Vertical Angle Theorem,

14. If ZY = 2x + 3 and WX = x + 4, find WX.

SOLUTION: Then ZP = 4(7) – 9 = 19. The opposite sides of a rectangle are parallel and Therefore, ZX = 2(ZP) = 38. congruent. So, ZY = WX. 2x + 3 = x + 4 ANSWER: x = 1 38 Use the value of x to find WX. WX = 1 + 4 = 5. 18. If , find . ANSWER: 5 SOLUTION: All four angles of a rectangle are right angles. So,

15. If PY = 3x – 5 and WP = 2x + 11, find ZP.

SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP.

ANSWER: The diagonals of a rectangle are congruent and 48 bisect each other. So, is an isosceles triangle. 19. If , find . SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 43 is a right triangle. Then,

16. If , find . SOLUTION: All four angles of a rectangle are right angles. So, Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: 46 PROOF Write a two-column proof. ANSWER: 20. Given: ABCD is a rectangle. 39 Prove: 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to Then ZP = 4(7) – 9 = 19. prove. Here, you are given ABCD is a rectangle. Therefore, ZX = 2(ZP) = 38. You need to prove . Use the properties that you have learned about rectangles to ANSWER: walk through the proof. 38 Given: ABCD is a rectangle. 18. If , find Prove: . Proof: Statements (Reasons) SOLUTION: 1. ABCD is a rectangle. (Given) All four angles of a rectangle are right angles. So, 2. ABCD is a parallelogram. (Def. of rectangle)

3. (Opp. sides of a .)

4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS) ANSWER: ANSWER: Proof: 48 Statements (Reasons) 1. ABCD is a rectangle. (Given) 19. If , find 2. ABCD is a parallelogram. (Def. of rectangle) . 3. (Opp. sides of a .) SOLUTION: 4. (Refl. Prop.) All four angles of a rectangle are right angles. So, 5. (Diagonals of a rectangle are .) is a right triangle. Then, 6. (SSS)

21. Given: QTVW is a rectangle.

Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46. Prove: ANSWER: SOLUTION: 46 You need to walk through the proof step by step. PROOF Write a two-column proof. Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and 20. Given: ABCD is a rectangle. Prove: . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: QTVW is a rectangle.

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle.

You need to prove . Use the Prove: properties that you have learned about rectangles to Proof: walk through the proof. Statements (Reasons)

Given: ABCD is a rectangle. 1. QTVW is a rectangle; .(Given) Prove: 2. QTVW is a parallelogram. (Def. of rectangle) Proof: 3. (Opp sides of a .) Statements (Reasons) 4. are right angles.(Def. of rectangle) 1. ABCD is a rectangle. (Given) 5. (All rt .) 2. ABCD is a parallelogram. (Def. of rectangle) 6. QR = ST (Def. of segs.) 3. (Opp. sides of a .) 7. (Refl. Prop.) 4. (Refl. Prop.) 8. RS = RS (Def. of segs.)

5. (Diagonals of a rectangle are .) 9. QR + RS = RS + ST (Add. prop.) 6. (SSS) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) ANSWER:

Proof: 12. (Def. of segs.) Statements (Reasons) 13. (SAS) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) ANSWER: 3. (Opp. sides of a .) Proof: Statements (Reasons) 4. (Refl. Prop.) 1. QTVW is a rectangle; .(Given)

5. (Diagonals of a rectangle are .) 2. QTVW is a parallelogram. (Def. of rectangle) 6. (SSS) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) Given: 21. QTVW is a rectangle. 5. (All rt .)

6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) Prove: 11. QS = RT (Subst.) 12. (Def. of segs.) SOLUTION: 13. (SAS) You need to walk through the proof step by step. Look over what you are given and what you need to COORDINATE GEOMETRY Graph each prove. Here, you are given QTVW is a rectangle and quadrilateral with the given vertices. Determine . You need to prove . Use whether the figure is a rectangle. Justify your the properties that you have learned about rectangles answer using the indicated formula. to walk through the proof. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula

Given: QTVW is a rectangle. SOLUTION:

Prove: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) Use the slope formula to find the slope of the sides of 3. (Opp sides of a .) the quadrilateral. 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS) The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. ANSWER: Therefore, the quadrilateral WXYZ is a parallelogram. Proof: The products of the slopes of the adjacent sides are Statements (Reasons) –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right 1. QTVW is a rectangle; .(Given) angles. Therefore, WXYZ is a rectangle. 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) ANSWER: 4. are right angles.(Def. of rectangle) Yes; slope of slope of , slope 5. (All rt .) 6. QR = ST (Def. of segs.) of slope of . So WXYZ is a parallelogram. The product of the slopes of 7. (Refl. Prop.) consecutive sides is –1, so the consecutive sides are 8. RS = RS (Def. of segs.) perpendicular and form right angles. Thus, WXYZ is a 9. QR + RS = RS + ST (Add. prop.) rectangle. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION: SOLUTION:

First, use the Distance Formula to find the lengths of First, Use the Distance formula to find the lengths of the sides of the quadrilateral. the sides of the quadrilateral.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION: 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, use the slope formula to find the lengths of the First, use the Distance Formula to find the lengths of sides of the quadrilateral. the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the The slopes of each pair of opposite sides are equal. lengths of the diagonals. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a parallelogram. So the diagonals are congruent. Thus, QRST is a None of the adjacent sides have slopes whose rectangle. product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle. ANSWER: ANSWER: Yes; so QRST is a parallelogram. so the diagonals are No; slope of slope of and slope of congruent. QRST is a rectangle. slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

26.

SOLUTION: First, use the slope formula to find the lengths of the All four angles of a rectangle are right angles. So, sides of the quadrilateral.

ANSWER: 50

No; slope of slope of and slope of 28. slope of . So, GHJK is a SOLUTION: parallelogram. The product of the slopes of The measures of angles 2 and 3 are equal as they consecutive sides ≠ –1, so the consecutive sides are are alternate interior angles. not perpendicular. Thus, GHJK is not a rectangle. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Quadrilateral ABCD is a rectangle. Find each Since the diagonals of a rectangle are congruent and measure if . bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: 80 26. SOLUTION: 30. All four angles of a rectangle are right angles. So, SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: The measures of angles 1 and 4 are equal as they 50 are alternate interior angles. Therefore, m∠4 = m∠1 = 50. 27. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 SOLUTION: and 6 is an isosceles triangle. Therefore, m∠6 = The measures of angles 2 and 3 are equal as they m∠4 = 50. are alternate interior angles. Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 3, 7 50 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. 31. ANSWER: SOLUTION: 40 The measures of angles 2 and 3 are equal as they are alternate interior angles. 28. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 SOLUTION: and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The measures of angles 2 and 3 are equal as they The sum of the three angles of a triangle is 180. are alternate interior angles. Therefore, m∠8 = 180 – (40 + 40) = 100. Therefore, m∠3 = m∠2 = 40. ANSWER: ANSWER: 100 40 32. CCSS MODELING Jody is building a new 29. bookshelf using wood and metal supports like the one SOLUTION: shown. To what length should she cut the metal supports in order for the bookshelf to be square, All four angles of a rectangle are right angles. So, which means that the angles formed by the shelves

and the vertical supports are all right angles? Explain your reasoning.

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50.

The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. SOLUTION: ANSWER: In order for the bookshelf to be square, the lengths of 80 all of the metal supports should be equal. Since we know the length of the bookshelves and the distance 30. between them, use the Pythagorean Theorem to find the lengths of the metal supports. SOLUTION: Let x be the length of each support. Express 3 feet All four angles of a rectangle are right angles. So, as 36 inches.

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Therefore, the metal supports should each be 39 Since the diagonals of a rectangle are congruent and inches long or 3 feet 3 inches long. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = ANSWER: m∠4 = 50. 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports ANSWER: should be equal. Since I knew the length of the 50 bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the 31. metal supports. SOLUTION: PROOF Write a two-column proof. The measures of angles 2 and 3 are equal as they 33. Theorem 6.13 are alternate interior angles. SOLUTION: Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 You need to walk through the proof step by step. Look over what you are given and what you need to and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. prove. Here, you are given WXYZ is a rectangle with Therefore, m∠8 = 180 – (40 + 40) = 100. diagonals . You need to prove . Use the properties that you have learned ANSWER: about rectangles to walk through the proof. 100 Given: WXYZ is a rectangle with diagonals 32. CCSS MODELING Jody is building a new

bookshelf using wood and metal supports like the one . shown. To what length should she cut the metal Prove: supports in order for the bookshelf to be square, which means that the angles formed by the shelves and the vertical supports are all right angles? Explain your reasoning.

Therefore, the metal supports should each be 39 inches long or 3 feet 3 inches long. Proof: ANSWER: 1. WXYZ is a rectangle with diagonals . 3 ft 3 in.; Sample answer: In order for the bookshelf (Given) to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the 2. (Opp. sides of a .) bookshelves and the distance between them, I used 3. (Refl. Prop.) the Pythagorean Theorem to find the lengths of the 4. are right angles. (Def. of

metal supports. rectangle) PROOF Write a two-column proof. 5. (All right .) 33. Theorem 6.13 6. (SAS)

SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. Look over what you are given and what you need to 34. Theorem 6.14 prove. Here, you are given WXYZ is a rectangle with SOLUTION: diagonals . You need to prove You need to walk through the proof step by step. . Use the properties that you have learned Look over what you are given and what you need to about rectangles to walk through the proof. prove. Here, you are given . You need to Given: WXYZ is a rectangle with diagonals prove that WXYZ is a rectangle.. Use the properties . that you have learned about rectangles to walk Prove: through the proof.

Given: Prove: WXYZ is a rectangle.

Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) Proof:

3. (Refl. Prop.) 1. (Given) 4. are right angles. (Def. of 2. (SSS) rectangle) 3. (CPCTC) 5. (All right .) 4. (Def. of ) 6. (SAS) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 7. (CPCTC) 6. are supplementary. (Cons. ANSWER: are suppl.) Given: WXYZ is a rectangle with diagonals 7. (Def of suppl.) . 8. are right angles. (If Prove: 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

ANSWER: Proof: Given: 1. WXYZ is a rectangle with diagonals . Prove: WXYZ is a rectangle. (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) Proof: 5. (All right .) 1. (Given) 6. (SAS) 2. (SSS) 7. (CPCTC) 3. (CPCTC) 4. (Def. of ) 34. Theorem 6.14 5. WXYZ is a parallelogram. (If both pairs of opp. SOLUTION: sides are , then quad. is .) You need to walk through the proof step by step. 6. are supplementary. (Cons. Look over what you are given and what you need to are suppl.) prove. Here, you are given 7. (Def of suppl.) . You need to 8. are right angles. (If prove that WXYZ is a rectangle.. Use the properties 2 and suppl., each is a rt. .) that you have learned about rectangles to walk through the proof. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) Given: Prove: WXYZ is a rectangle. PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION: You need to walk through the proof step by step.

Proof: Look over what you are given and what you need to 1. (Given) prove. Here, you are given a parallelogram with one 2. (SSS) right angle. You need to prove that it is a rectangle. 3. (CPCTC) Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk

4. (Def. of ) through the proof. 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If

2 and suppl., each is a rt. .) ABCD is a parallelogram, and is a right angle. 9. are right angles. (If a has 1 Since ABCD is a parallelogram and has one right rt. , it has 4 rt. .) angle, then it has four right angles. So by the 10. WXYZ is a rectangle. (Def. of rectangle) definition of a rectangle, ABCD is a rectangle.

ANSWER: ANSWER: Given: Prove: WXYZ is a rectangle.

ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right Proof: angle, then it has four right angles. So by the 1. (Given) definition of a rectangle, ABCD is a rectangle. 2. (SSS) 3. (CPCTC) 36. If a quadrilateral has four right angles, then it is a rectangle. 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. SOLUTION: sides are , then quad. is .) You need to walk through the proof step by step. 6. are supplementary. (Cons. Look over what you are given and what you need to are suppl.) prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. 7. (Def of suppl.) Let ABCD be a quadrilateral with four right angles. 8. are right angles. (If Then use the properties of parallelograms and 2 and suppl., each is a rt. .) rectangles to walk through the proof. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

Step 2: Place the compass on point P. Using the ABCD is a parallelogram, and is a right angle. same compass setting, draw an arc to the left and Since ABCD is a parallelogram and has one right right of P that intersects . angle, then it has four right angles. So by the 6-4 Rectangles definition of a rectangle, ABCD is a rectangle. Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. 36. If a quadrilateral has four right angles, then it is a Place the compass on each point of intersection and rectangle. draw arcs that intersect above . SOLUTION: You need to walk through the proof step by step. Step 4: Draw line a through point P and the Look over what you are given and what you need to intersection of the arcs drawn in step 3. prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Step 5: Repeat steps 2 and 3 using point Q. Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and Step 6: Draw line b through point Q and the rectangles to walk through the proof. intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that

ABCD is a quadrilateral with four right angles. intersects line b. Label the point of intersection S. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 9: Draw .

ANSWER:

Step 2: Place the compass on point P. Using the ANSWER: same compass setting, draw an arc to the left and right of P that intersects . Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that Step 3: Open the compass to a setting greater than are perpendicular to the same line are parallel, so the distance from P to either point of intersection. . The same compass setting was used to Place the compass on each point of intersection and eSolutions Manual - Powered by Cognero locate points R and S, so . If one pair Pageof 13 draw arcs that intersect above . opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 4: Draw line a through point P and the A parallelogram with right angles is a rectangle. intersection of the arcs drawn in step 3. Thus, PRSQ is a rectangle.

Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Since and , the measure of ANSWER: angles P and Q is 90 degrees. Lines that are Sample answer: He should measure the diagonals of perpendicular to the same line are parallel, so the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are . The same compass setting was used to congruent, then the end zone is a rectangle. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and ALGEBRA Quadrilateral WXYZ is a rectangle. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

ANSWER: 39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: Sample answer: Since and , the The diagonals of a rectangle are congruent to each measure of angles P and Q is 90 degrees. Lines that other. So, YW = XZ = b. All four angles of a are perpendicular to the same line are parallel, so rectangle are right angles. So, is a right 2 2 . The same compass setting was used to triangle. By the Pythagorean Theorem, XZ = XW + 2 WZ . locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Therefore, YW = 5. Thus, PRSQ is a rectangle. ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: The diagonals of a rectangle are congruent to each other. So, WY = XZ = 2c. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XY2 + 2 ZY . 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. Therefore, WY = 2(5) = 10. SOLUTION: Sample answer: If the diagonals of a parallelogram ANSWER: are congruent, then it is a rectangle. He should 10 measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides 41. SIGNS The sign below is in the foyer of Nyoko’s are congruent, then the quadrilateral is a school. Based on the dimensions given, can Nyoko parallelogram. If the diagonals are congruent, then be sure that the sign is a rectangle? Explain your the end zone is a rectangle. reasoning. ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle. SOLUTION:

Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that 39. If XW = 3, WZ = 4, and XZ = b, find YW. can be used to prove that it is a rectangle. If the SOLUTION: diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we The diagonals of a rectangle are congruent to each could prove that the sign was a rectangle. other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right ANSWER: triangle. By the Pythagorean Theorem, XZ2 = XW 2 + No; sample answer: Both pairs of opposite sides are 2 WZ . congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

PROOF Write a coordinate proof of each Therefore, YW = 5. statement. ANSWER: 42. The diagonals of a rectangle are congruent. 5 SOLUTION: Begin by positioning quadrilateral ABCD on a 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b SOLUTION: units. Then the rest of the vertices are A(0, b), B(a, The diagonals of a rectangle are congruent to each b), and C(a, 0). You need to walk through the proof other. So, WY = XZ = 2c. All four angles of a step by step. Look over what you are given and what rectangle are right angles. So, is a right you need to prove. Here, you are given ABCD is a triangle. By the Pythagorean Theorem, XZ2 = XY2 + rectangle and you need to prove . Use the 2 ZY . properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove:

Therefore, WY = 2(5) = 10.

ANSWER: 10

The diagonals, have the same length, so they are congruent.

ANSWER: No; sample answer: Both pairs of opposite sides are Proof: congruent, so the sign is a parallelogram, but no Use the Distance Formula to find measure is given that can be used to prove that it is a and . rectangle. have the same length, so they are PROOF Write a coordinate proof of each congruent. statement. 42. The diagonals of a rectangle are congruent. 43. If the diagonals of a parallelogram are congruent, SOLUTION: then it is a rectangle. Begin by positioning quadrilateral ABCD on a SOLUTION: coordinate plane. Place vertex D at the origin. Let Begin by positioning parallelogram ABCD on a the length of the bases be a units and the height be b coordinate plane. Place vertex A at the origin. Let units. Then the rest of the vertices are A(0, b), B(a, the length of the bases be a units and the height be c b), and C(a, 0). You need to walk through the proof units. Then the rest of the vertices are B(a, 0), C(b + step by step. Look over what you are given and what a, c), and D(b, c). You need to walk through the you need to prove. Here, you are given ABCD is a proof step by step. Look over what you are given rectangle and you need to prove . Use the and what you need to prove. Here, you are given properties that you have learned about rectangles to ABCD and and you need to prove that walk through the proof. ABCD is a rectangle. Use the properties that you

have learned about rectangles to walk through the Given: ABCD is a rectangle. proof. Prove: Given: ABCD and Prove: ABCD is a rectangle.

Proof: Use the Distance Formula to find the lengths of the Proof: diagonals.

The diagonals, have the same length, so AC = BD. So, they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.

Proof: ANSWER:

Use the Distance Formula to find Given: ABCD and and . Prove: ABCD is a rectangle. have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION: Proof: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b +

a, c), and D(b, c). You need to walk through the But AC = BD. So, proof step by step. Look over what you are given

and what you need to prove. Here, you are given

ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Because A and B are different points, Prove: ABCD is a rectangle. a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. Proof: a. GEOMETRIC Draw three parallelograms, each

with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. AC = BD. So, b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

ANSWER:

Given: ABCD and Prove: ABCD is a rectangle.

b. Use a protractor to measure each angle listed in the table. Proof:

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each

But AC = BD. So, formed by diagonals. The diagonals of a parallelogram with four congruent sides are

perpendicular.

ANSWER: a. Sample answer:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one b. parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. c. Sample answer: The diagonals of a parallelogram b. TABULAR Use a protractor to measure the with four congruent sides are perpendicular. appropriate angles and complete the table below. 45. CHALLENGE In rectangle ABCD, Find the c. VERBAL Make a conjecture about the diagonals values of x and y. of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and b. Use a protractor to measure each angle listed in bisect each other. So, is an isosceles triangle the table. with So,

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are The sum of the angles of a triangle is 180. perpendicular. So, ANSWER: By the Vertical Angle Theorem, a. Sample answer:

ANSWER: x = 6, y = −10

b. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a c. Sample answer: The diagonals of a parallelogram single vertex of a triangle. with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the

values of x and y. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

SOLUTION: So, Tamika is correct. All four angles of a rectangle are right angles. So, ANSWER: Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of The diagonals of a rectangle are congruent and the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of bisect each other. So, is an isosceles triangle the angles in the congruent triangles has to be a right with angle. So, 47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines? The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

So, Tamika is correct. A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that ANSWER: run parallel to the x-axis will be parallel to each other Tamika; Sample answer: When two congruent and perpendicular to lines parallel to the y-axis. triangles are arranged to form a quadrilateral, two of Selecting y = 0 and x = 0 will form 2 sides of the the angles are formed by a single vertex of a triangle. rectangle. The lines y = 4 and x = 6 form the other 2 In order for the quadrilateral to be a rectangle, one of sides. the angles in the congruent triangles has to be a right angle. x = 0, x = 6, y = 0, y = 4;

47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines?

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by SOLUTION: Theorem 6.12, the quadrilateral is a parallelogram. Let the lines n, p, q, and r intersect the line l at the Since is horizontal and is vertical, the lines points A, B, C, and D respectively. Similarly, let the are perpendicular and the measure of the angle they lines intersect the line m at the points W, X, Y, and Z. form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6 The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the 48. OPEN ENDED Write the equations of four lines slope of DC is 0. Since a pair of sides of the having intersections that form the vertices of a quadrilateral is both parallel and congruent, by rectangle. Verify your answer using coordinate Theorem 6.12, the quadrilateral is a parallelogram. geometry. Since is horizontal and is vertical, the lines SOLUTION: are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has Sample answer: one right angle, it has four right angles. Therefore, by A rectangle has 4 right angles and each pair of definition, the parallelogram is a rectangle. opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other 49. WRITING IN MATH Explain why all rectangles and perpendicular to lines parallel to the y-axis. are parallelograms, but all parallelograms are not Selecting y = 0 and x = 0 will form 2 sides of the rectangles. rectangle. The lines y = 4 and x = 6 form the other 2 sides. SOLUTION: Sample answer: A parallelogram is a quadrilateral in x = 0, x = 6, y = 0, y = 4; which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms.

Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

ANSWER: Sample answer: All rectangles are parallelograms The length of is 6 − 0 or 6 units and the length of because, by definition, both pairs of opposite sides is 6 – 0 or 6 units. The slope of AB is 0 and the are parallel. Parallelograms with right angles are slope of DC is 0. Since a pair of sides of the rectangles, so some parallelograms are rectangles, quadrilateral is both parallel and congruent, by but others with non-right angles are not. Theorem 6.12, the quadrilateral is a parallelogram. 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = Since is horizontal and is vertical, the lines 13, what values of x and y make parallelogram are perpendicular and the measure of the angle they FGHJ a rectangle? form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER:

Sample answer: x = 0, x = 6, y = 0, A x = 3, y = 4 y = 4; B x = 4, y = 3 C x = 7, y = 8 D x = 8, y = 7 SOLUTION: The opposite sides of a parallelogram are congruent. So, FJ = GH. –3x + 5y = 11 FJ = GH

If the diagonals of a parallelogram are congruent, The length of is 6 − 0 or 6 units and the length of then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a is 6 – 0 or 6 units. The slope of AB is 0 and the rectangle, then will be an isosceles triangle slope of DC is 0. Since a pair of sides of the with That is, 3x + y = 13. quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Add the two equations to eliminate the x-term and Since is horizontal and is vertical, the lines solve for y. are perpendicular and the measure of the angle they –3x + 5y = 11 form is 90. By Theorem 6.6, if a parallelogram has 3x + y = 13 one right angle, it has four right angles. Therefore, by 6y = 24 Add to eliminate x-term. definition, the parallelogram is a rectangle. y = 4 Divide each side by 6.

49. WRITING IN MATH Explain why all rectangles Use the value of y to find the value of x. are parallelograms, but all parallelograms are not 3x + 4 = 13 Substitute. rectangles. 3x = 9 Subtract 4 from each side. SOLUTION: x = 3 Divide each side by 3. Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By So, x = 3, y = 4, the correct choice is A. definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always ANSWER: parallelograms. A

If the diagonals of a parallelogram are congruent, SOLUTION: then it is a rectangle. Since FGHJ is a parallelogram, The diagonals of a rectangle are congruent and the diagonals bisect each other. So, if it is a bisect each other. So, is an isosceles triangle rectangle, then will be an isosceles triangle with with That is, 3x + y = 13. So, By the Exterior Angle Theorem, Add the two equations to eliminate the x-term and solve for y. –3x + 5y = 11 ANSWER: 3x + y = 13 112 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6. 53. SAT/ACT If p is odd, which of the following must also be odd? Use the value of y to find the value of x. A 2p 3x + 4 = 13 Substitute. B 2p + 2 3x = 9 Subtract 4 from each side. x = 3 Divide each side by 3. C D So, x = 3, y = 4, the correct choice is A. E p + 2 ANSWER: SOLUTION: A Analyze each answer choice to determine which expression is odd when p is odd. 51. ALGEBRA A rectangular playground is surrounded Choice A 2p by an 80-foot fence. One side of the playground is 10 This is a multiple of 2 so it is even. feet longer than the other. Which of the following Choice B 2p + 2 equations could be used to find r, the shorter side of the playground? This can be rewritten as 2(p + 1) which is also a F 10r + r = 80 multiple of 2 so it is even.

G 4r + 10 = 80 Choice C H r(r + 10) = 80 J 2(r + 10) + 2r = 80 If p is odd , is not a whole number. Choice D 2p – 2 SOLUTION: This can be rewritten as 2(p - 1) which is also a The formula to find the perimeter of a rectangle is multiple of 2 so it is even. given by P = 2(l + w). The perimeter is given to be 8 Choice E p + 2 ft. The width is r and the length is r + 10. So, 2r + 2 Adding 2 to an odd number results in an odd (r + 10) = 80. number. This is odd. Therefore, the correct choice is J. The correct choice is E. ANSWER: ANSWER: J E 52. SHORT RESPONSE What is the measure of ALGEBRA Find x and y so that the

? quadrilateral is a parallelogram.

ANSWER: 112

53. SAT/ACT If p is odd, which of the following must also be odd? A 2p ANSWER: B 2p + 2 x = 2, y = 41

C D E p + 2 SOLUTION: Analyze each answer choice to determine which 55. expression is odd when p is odd. Choice A 2p SOLUTION: This is a multiple of 2 so it is even. The opposite sides of a parallelogram are congruent. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

ALGEBRA Find x and y so that the quadrilateral is a parallelogram. 56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 54. 2y = 14 SOLUTION: y = 7 The opposite sides of a parallelogram are congruent. Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x 2 = x In a parallelogram, consecutive interior angles are supplementary. ANSWER:

x = 2, y = 7

Therefore, the coordinates of the intersection of the 55. diagonals are (2.5, 0.5).

SOLUTION: ANSWER: The opposite sides of a parallelogram are congruent. (2.5, 0.5)

Refer to the figure.

The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

58. If , name two congruent angles. ANSWER: SOLUTION: x = 8, y = 22 Since is an isosceles triangle. So,

ANSWER:

56. SOLUTION: 59. If , name two congruent segments. The diagonals of a parallelogram bisect each other. SOLUTION: So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. In a triangle with two congruent angles, the sides 2y = 14 opposite to the congruent angles are also congruent. y = 7 So, . Use the value of y in the second equation and solve for x. ANSWER: 7 + 3 = 2x + 6 4 = 2x 2 = x 60. If , name two congruent segments. ANSWER: SOLUTION: x = 2, y = 7 In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. 57. COORDINATE GEOMETRY Find the So, . coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C ANSWER: (4, –2), and D(–1, –1). SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps ANSWER: to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure. 3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Therefore,

ANSWER: 1. QR 33.5 SOLUTION: 4. The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft. SOLUTION: The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 7 ft Then, By the Vertical Angle Theorem, 2. SQ SOLUTION: Therefore, The diagonals of a rectangle bisect each other. So, ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

ANSWER: 5. If FD = 3x – 7 and EG = x + 5, find EG.

SOLUTION: The diagonals of a rectangle are congruent to each 3. other. So, FD = EG.

SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Use the value of x to find EG. Then, EG = 6 + 5 = 11 By the Exterior Angle Theorem, ANSWER: 11 Therefore, 6. If , ANSWER: find . 33.5 SOLUTION: 4. All four angles of a rectangle are right angles. So, SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore, ANSWER: 51 ANSWER: 56.5 7. PROOF If ABDE is a rectangle and , ALGEBRA Quadrilateral DEFG is a rectangle. prove that .

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each SOLUTION: other. So, FD = EG. You need to walk through the proof step by step. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and . You need to prove . Use the Use the value of x to find EG. properties that you have learned about rectangles to EG = 6 + 5 = 11 walk through the proof.

ANSWER: Given: ABDE is a rectangle; 11 Prove:

6. If , find . SOLUTION: All four angles of a rectangle are right angles. So,

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) ANSWER: 4. are right angles.(Def. of rectangle) 51 5. (All rt .) 6. (SAS) 7. PROOF If ABDE is a rectangle and , 7. (CPCTC) prove that . ANSWER: Given: ABDE is a rectangle; Prove:

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to Statements(Reasons) prove. You are given ABDE is a rectangle and 1. ABDE is a rectangle; . . You need to prove . Use the (Given) properties that you have learned about rectangles to 2. ABDE is a parallelogram. (Def. of walk through the proof. rectangle)

3. (Opp. sides of a .) Given: ABDE is a rectangle; 4. are right angles.(Def. of rectangle) Prove: 5. (All rt .) 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your Statements(Reasons) answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula 1. ABDE is a rectangle; . (Given) SOLUTION: 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle;

Prove: Use the slope formula to find the slope of the sides of the quadrilateral.

Statements(Reasons) 1. ABDE is a rectangle; . (Given)

2. ABDE is a parallelogram. (Def. of There are no parallel sides, so WXYZ is not a

rectangle) parallelogram. Therefore, WXYZ is not a rectangle. 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) ANSWER: 5. (All rt .) No; slope of , slope of , slope of 6. (SAS) 7. (CPCTC) , and slope of . Slope of

COORDINATE GEOMETRY Graph each slope of , and slope of slope of quadrilateral with the given vertices. Determine , so WXYZ is not a parallelogram. Therefore, whether the figure is a rectangle. Justify your WXYZ is not a rectangle. answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula Use the slope formula to find the slope of the sides of SOLUTION: the quadrilateral.

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. First, Use the Distance formula to find the lengths of ANSWER: the sides of the quadrilateral.

No; slope of , slope of , slope of

, and slope of . Slope of

slope of , and slope of slope of so ABCD is a parallelogram. , so WXYZ is not a parallelogram. Therefore, A parallelogram is a rectangle if the diagonals are WXYZ is not a rectangle. congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are 10. BC congruent. Use the Distance formula to find the lengths of the diagonals. SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

So the diagonals are congruent. Thus, ABCD is a ANSWER: rectangle. 2 ft ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a 11. DB parallelogram. , so the diagonals are SOLUTION: congruent. Thus, ABCD is a rectangle. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

ANSWER: 6.3 ft

12. FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, SOLUTION: AD = 2 feet, and , find each The diagonals of a rectangle are congruent and measure. bisect each other. So, is an isosceles triangle. Then,

By the Vertical Angle Theorem,

SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . ANSWER: BC = AD = 2 25

DC = AB = 6 CCSS REGULARITY Quadrilateral WXYZ is a

rectangle.

ANSWER: 6.3 ft

12. 14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and The opposite sides of a rectangle are parallel and bisect each other. So, is an isosceles congruent. So, ZY = WX. triangle. 2x + 3 = x + 4 Then, x = 1 Use the value of x to find WX. WX = 1 + 4 = 5.

By the Vertical Angle Theorem, ANSWER: 5

ANSWER: 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. 50 SOLUTION: 13. The diagonals of a rectangle bisect each other. So, PY = WP. SOLUTION:

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles The diagonals of a rectangle are congruent and triangle. bisect each other. So, is an isosceles triangle. Then, All four angles of a rectangle are right angles.

ANSWER: ANSWER: 43 25 16. If , find CCSS REGULARITY Quadrilateral WXYZ is a . rectangle. SOLUTION: All four angles of a rectangle are right angles. So,

14. If ZY = 2x + 3 and WX = x + 4, find WX.

SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. ANSWER: 2x + 3 = x + 4 39 x = 1 Use the value of x to find WX. 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. WX = 1 + 4 = 5. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 5 bisect each other. So, ZP = PY.

15. If PY = 3x – 5 and WP = 2x + 11, find ZP.

SOLUTION: Then ZP = 4(7) – 9 = 19. The diagonals of a rectangle bisect each other. So, Therefore, ZX = 2(ZP) = 38. PY = WP. ANSWER: 38 The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 18. If , find . SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 43

16. If , find .

SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 48

19. If , find . SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. Then,

ANSWER: 39 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: Therefore, Since is a The diagonals of a rectangle are congruent and transversal of parallel sides and , alternate bisect each other. So, ZP = PY. interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: 46 Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38. PROOF Write a two-column proof. ANSWER: 20. Given: ABCD is a rectangle. Prove: 38

18. If , find . SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

ANSWER: Given: ABCD is a rectangle. 48 Prove:

19. If , find Proof: Statements (Reasons) . 1. ABCD is a rectangle. (Given) SOLUTION: 2. ABCD is a parallelogram. (Def. of rectangle) All four angles of a rectangle are right angles. So, 3. (Opp. sides of a .) is a right triangle. Then, 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS)

ANSWER: Proof: Therefore, Since is a Statements (Reasons) transversal of parallel sides and , alternate 1. ABCD is a rectangle. (Given) interior angles WZX and ZXY are congruent. So, 2. ABCD is a parallelogram. (Def. of rectangle) = = 46. 3. (Opp. sides of a .)

ANSWER: 4. (Refl. Prop.) 46 5. (Diagonals of a rectangle are .) 6. (SSS) PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. 21. Given: QTVW is a rectangle. Prove:

SOLUTION: Prove: You need to walk through the proof step by step. Look over what you are given and what you need to SOLUTION: prove. Here, you are given ABCD is a rectangle. You need to walk through the proof step by step. You need to prove . Use the Look over what you are given and what you need to properties that you have learned about rectangles to prove. Here, you are given QTVW is a rectangle and walk through the proof. . You need to prove . Use the properties that you have learned about rectangles Given: ABCD is a rectangle. to walk through the proof. Prove: Proof: Given: QTVW is a rectangle. Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) Prove: 6. (SSS) Proof: ANSWER: Statements (Reasons) Proof: 1. QTVW is a rectangle; .(Given) Statements (Reasons) 2. QTVW is a parallelogram. (Def. of rectangle) 1. ABCD is a rectangle. (Given) 3. (Opp sides of a .) 2. ABCD is a parallelogram. (Def. of rectangle) 4. are right angles.(Def. of rectangle) 3. (Opp. sides of a .) 5. (All rt .)

4. (Refl. Prop.) 6. QR = ST (Def. of segs.)

5. (Diagonals of a rectangle are .) 7. (Refl. Prop.)

6. (SSS) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 21. Given: QTVW is a rectangle. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Prove: Statements (Reasons)

SOLUTION: 1. QTVW is a rectangle; .(Given) You need to walk through the proof step by step. 2. QTVW is a parallelogram. (Def. of rectangle) Look over what you are given and what you need to 3. (Opp sides of a .) prove. Here, you are given QTVW is a rectangle and 4. are right angles.(Def. of rectangle) . You need to prove . Use 5. (All rt .) the properties that you have learned about rectangles 6. QR = ST (Def. of segs.)

to walk through the proof. 7. (Refl. Prop.) Given: QTVW is a rectangle. 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

Prove: COORDINATE GEOMETRY Graph each Proof: quadrilateral with the given vertices. Determine Statements (Reasons) whether the figure is a rectangle. Justify your 1. QTVW is a rectangle; .(Given) answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) SOLUTION: 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS) Use the slope formula to find the slope of the sides of ANSWER: the quadrilateral. Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) The slopes of each pair of opposite sides are equal. 8. RS = RS (Def. of segs.) So, the two pairs of opposite sides are parallel. 9. QR + RS = RS + ST (Add. prop.) Therefore, the quadrilateral WXYZ is a parallelogram. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) The products of the slopes of the adjacent sides are 11. QS = RT (Subst.) –1. So, any two adjacent sides are perpendicular to 12. (Def. of segs.) each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle. 13. (SAS) ANSWER: COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine Yes; slope of slope of , slope whether the figure is a rectangle. Justify your answer using the indicated formula. of slope of . So WXYZ is a 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are SOLUTION: perpendicular and form right angles. Thus, WXYZ is a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a so JKLM is a rectangle. parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance rectangle. Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION: First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the

lengths of the diagonals. First, use the Distance Formula to find the lengths of the sides of the quadrilateral. So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER:

No; so JKLM is a so QRST is a parallelogram; so parallelogram. the diagonals are not congruent. Thus, JKLM is not a A parallelogram is a rectangle if the diagonals are rectangle. congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals. First, use the slope formula to find the lengths of the sides of the quadrilateral.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

ANSWER:

No; slope of slope of and slope of

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula slope of . So, GHJK is a SOLUTION: parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

First, use the slope formula to find the lengths of the sides of the quadrilateral. Quadrilateral ABCD is a rectangle. Find each measure if .

26. The slopes of each pair of opposite sides are equal. SOLUTION: So, the two pairs of opposite sides are parallel. All four angles of a rectangle are right angles. So, Therefore, the quadrilateral GHJK is a parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle. ANSWER: ANSWER: 50 No; slope of slope of and slope of 27. slope of . So, GHJK is a SOLUTION: parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are The measures of angles 2 and 3 are equal as they not perpendicular. Thus, GHJK is not a rectangle. are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: 40

28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. Quadrilateral ABCD is a rectangle. Find each ANSWER: measure if . 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

26. SOLUTION: All four angles of a rectangle are right angles. So, The measures of angles 1 and 4 are equal as they

are alternate interior angles. Therefore, m∠4 = m∠1 = 50.

Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5

ANSWER: and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. 50 Therefore, m∠5 = 180 – (50 + 50) = 80.

27. ANSWER: 80 SOLUTION: The measures of angles 2 and 3 are equal as they 30. are alternate interior angles. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 All four angles of a rectangle are right angles. So, and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40.

ANSWER: The measures of angles 1 and 4 are equal as they 40 are alternate interior angles. Therefore, m∠4 = m∠1 = 50.

28. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 The measures of angles 2 and 3 are equal as they and 6 is an isosceles triangle. Therefore, m∠6 = are alternate interior angles. m∠4 = 50. Therefore, m∠3 = m∠2 = 40. ANSWER: ANSWER: 50 40 31. 29. SOLUTION: SOLUTION: The measures of angles 2 and 3 are equal as they All four angles of a rectangle are right angles. So, are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7

and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. The measures of angles 1 and 4 are equal as they Therefore, m∠8 = 180 – (40 + 40) = 100. are alternate interior angles. Therefore, m∠4 = m∠1 = 50. ANSWER: Since the diagonals of a rectangle are congruent and 100 bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. 32. CCSS MODELING Jody is building a new The sum of the three angles of a triangle is 180. bookshelf using wood and metal supports like the one Therefore, m∠5 = 180 – (50 + 50) = 80. shown. To what length should she cut the metal supports in order for the bookshelf to be square, ANSWER: which means that the angles formed by the shelves 80 and the vertical supports are all right angles? Explain your reasoning. 30. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they

are alternate interior angles. Therefore, m∠4 = m∠1 = 50. SOLUTION: Since the diagonals of a rectangle are congruent and In order for the bookshelf to be square, the lengths of bisect each other, the triangle with the angles 4, 5 all of the metal supports should be equal. Since we and 6 is an isosceles triangle. Therefore, m∠6 = know the length of the bookshelves and the distance m∠4 = 50. between them, use the Pythagorean Theorem to find the lengths of the metal supports. ANSWER: Let x be the length of each support. Express 3 feet 50 as 36 inches.

31. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, the metal supports should each be 39 Since the diagonals of a rectangle are congruent and inches long or 3 feet 3 inches long. bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. ANSWER: The sum of the three angles of a triangle is 180. 3 ft 3 in.; Sample answer: In order for the bookshelf Therefore, m∠8 = 180 – (40 + 40) = 100. to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the ANSWER: bookshelves and the distance between them, I used 100 the Pythagorean Theorem to find the lengths of the metal supports. 32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one PROOF Write a two-column proof. shown. To what length should she cut the metal 33. Theorem 6.13 supports in order for the bookshelf to be square, which means that the angles formed by the shelves SOLUTION: and the vertical supports are all right angles? Explain You need to walk through the proof step by step. your reasoning. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: WXYZ is a rectangle with diagonals .

Prove: SOLUTION: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find

the lengths of the metal supports. Proof: Let x be the length of each support. Express 3 feet as 36 inches. 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) Therefore, the metal supports should each be 39

inches long or 3 feet 3 inches long. 5. (All right .) 6. (SAS) ANSWER: 7. (CPCTC) 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports ANSWER: should be equal. Since I knew the length of the Given: WXYZ is a rectangle with diagonals bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the . metal supports. Prove:

PROOF Write a two-column proof. 33. Theorem 6.13 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to Proof: prove. Here, you are given WXYZ is a rectangle with 1. WXYZ is a rectangle with diagonals . diagonals . You need to prove (Given)

. Use the properties that you have learned 2. (Opp. sides of a .) about rectangles to walk through the proof. 3. (Refl. Prop.) 4. are right angles. (Def. of Given: WXYZ is a rectangle with diagonals rectangle) . 5. (All right .) Prove: 6. (SAS) 7. (CPCTC)

34. Theorem 6.14 SOLUTION: Proof: You need to walk through the proof step by step. 1. WXYZ is a rectangle with diagonals . Look over what you are given and what you need to (Given) prove. Here, you are given 2. (Opp. sides of a .) . You need to 3. (Refl. Prop.) prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk 4. are right angles. (Def. of through the proof.

rectangle) 5. (All right .) Given: 6. (SAS) Prove: WXYZ is a rectangle. 7. (CPCTC)

ANSWER: Given: WXYZ is a rectangle with diagonals

. Prove: Proof: 1. (Given) 2. (SSS) 3. (CPCTC) 4. (Def. of )

Proof: 5. WXYZ is a parallelogram. (If both pairs of opp.

1. WXYZ is a rectangle with diagonals . sides are , then quad. is .) (Given) 6. are supplementary. (Cons. are suppl.) 2. (Opp. sides of a .) 7. (Def of suppl.) 3. (Refl. Prop.) 8. are right angles. (If 4. are right angles. (Def. of 2 and suppl., each is a rt. .) rectangle) 5. (All right .) 9. are right angles. (If a has 1

6. (SAS) rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) 7. (CPCTC) ANSWER: 34. Theorem 6.14 Given: Prove: WXYZ is a rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given . You need to

prove that WXYZ is a rectangle.. Use the properties Proof: that you have learned about rectangles to walk through the proof. 1. (Given) 2. (SSS) Given: 3. (CPCTC) Prove: WXYZ is a rectangle. 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) Proof: 8. are right angles. (If

1. (Given) 2 and suppl., each is a rt. .)

2. (SSS) 9. are right angles. (If a has 1

3. (CPCTC) rt. , it has 4 rt. .) 4. (Def. of ) 10. WXYZ is a rectangle. (Def. of rectangle) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) PROOF Write a paragraph proof of each 6. are supplementary. (Cons. statement. 35. If a parallelogram has one right angle, then it is a

are suppl.) rectangle. 7. (Def of suppl.) SOLUTION: 8. are right angles. (If You need to walk through the proof step by step. 2 and suppl., each is a rt. .) Look over what you are given and what you need to 9. are right angles. (If a has 1 prove. Here, you are given a parallelogram with one rt. , it has 4 rt. .) right angle. You need to prove that it is a rectangle. 10. WXYZ is a rectangle. (Def. of rectangle) Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk ANSWER: through the proof. Given: Prove: WXYZ is a rectangle.

ABCD is a parallelogram, and is a right angle. Proof: Since ABCD is a parallelogram and has one right 1. (Given) angle, then it has four right angles. So by the 2. (SSS) definition of a rectangle, ABCD is a rectangle. 3. (CPCTC) ANSWER: 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) ABCD is a parallelogram, and is a right angle. 8. are right angles. (If Since ABCD is a parallelogram and has one right 2 and suppl., each is a rt. .) angle, then it has four right angles. So by the 9. are right angles. (If a has 1 definition of a rectangle, ABCD is a rectangle. rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) 36. If a quadrilateral has four right angles, then it is a rectangle. PROOF Write a paragraph proof of each statement. SOLUTION: 35. If a parallelogram has one right angle, then it is a You need to walk through the proof step by step. rectangle. Look over what you are given and what you need to prove. Here, you are given a quadrilateral with four SOLUTION: right angles. You need to prove that it is a rectangle. You need to walk through the proof step by step. Let ABCD be a quadrilateral with four right angles. Look over what you are given and what you need to Then use the properties of parallelograms and prove. Here, you are given a parallelogram with one rectangles to walk through the proof. right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

ANSWER:

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right 37. CONSTRUCTION Construct a rectangle using the angle, then it has four right angles. So by the construction for congruent segments and the definition of a rectangle, ABCD is a rectangle. construction for a line perpendicular to another line through a point on the line. Justify each step of the 36. If a quadrilateral has four right angles, then it is a construction. rectangle. SOLUTION: SOLUTION: Sample answer: You need to walk through the proof step by step. Step 1: Graph a line and place points P and Q on the Look over what you are given and what you need to line. prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Step 2: Place the compass on point P. Using the Let ABCD be a quadrilateral with four right angles. same compass setting, draw an arc to the left and Then use the properties of parallelograms and right of P that intersects .

rectangles to walk through the proof.

Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Place the compass on each point of intersection and draw arcs that intersect above .

Step 4: Draw line a through point P and the ABCD is a quadrilateral with four right angles. intersection of the arcs drawn in step 3. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a Step 5: Repeat steps 2 and 3 using point Q. rectangle, ABCD is a rectangle. ANSWER: Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

ABCD is a quadrilateral with four right angles. Step 8: Using the same compass setting, put the ABCD is a parallelogram because both pairs of compass at Q and draw an arc above it that opposite angles are congruent. By definition of a intersects line b. Label the point of intersection S. rectangle, ABCD is a rectangle. Step 9: Draw . 37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. SOLUTION: Sample answer: Step 1: Graph a line and place points P and Q on the line.

Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and Since and , the measure of right of P that intersects . angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 3: Open the compass to a setting greater than . The same compass setting was used to the distance from P to either point of intersection. locate points R and S, so . If one pair of Place the compass on each point of intersection and opposite sides of a quadrilateral is both parallel and draw arcs that intersect above . congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Step 4: Draw line a through point P and the Thus, PRSQ is a rectangle. intersection of the arcs drawn in step 3.

ANSWER: Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the Sample answer: Since and , the intersection of the arcs drawn in step 5. measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 7: Using any compass setting, put the compass . The same compass setting was used to at P and draw an arc above it that intersects line a. Label the point of intersection R. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and Step 8: Using the same compass setting, put the congruent, then the quadrilateral is a parallelogram. compass at Q and draw an arc above it that A parallelogram with right angles is a rectangle. intersects line b. Label the point of intersection S. Thus, PRSQ is a rectangle.

Step 9: Draw .

Thus, PRSQ is a rectangle. are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then the end zone is a rectangle. ANSWER: ANSWER: Sample answer: Since and , the Sample answer: He should measure the diagonals of measure of angles P and Q is 90 degrees. Lines that the end zone and both pairs of opposite sides. If the 6-4 Rectanglesare perpendicular to the same line are parallel, so diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. . The same compass setting was used to locate points R and S, so . If one pair of ALGEBRA Quadrilateral WXYZ is a rectangle. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

Therefore, YW = 5. 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for ANSWER: painting the field. He has finished the end zone. 5 Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: SOLUTION: Sample answer: If the diagonals of a parallelogram The diagonals of a rectangle are congruent to each are congruent, then it is a rectangle. He should other. So, WY = XZ = 2c. All four angles of a measure the diagonals of the end zone and both rectangle are right angles. So, is a right pairs of opposite sides. If both pairs of opposite sides triangle. By the Pythagorean Theorem, XZ2 = XY2 + are congruent, then the quadrilateral is a 2 parallelogram. If the diagonals are congruent, then ZY . the end zone is a rectangle.

ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are

congruent, then the end zone is a rectangle. Therefore, WY = 2(5) = 10. ANSWER: ALGEBRA Quadrilateral WXYZ is a rectangle. 10

41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your 39. If XW = 3, WZ = 4, and XZ = b, find YW. reasoning. SOLUTION: The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 eSolutionsWZ Manual. - Powered by Cognero Page 14

SOLUTION:

Therefore, YW = 5. Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that ANSWER: can be used to prove that it is a rectangle. If the 5 diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. could prove that the sign was a rectangle. SOLUTION: ANSWER: The diagonals of a rectangle are congruent to each No; sample answer: Both pairs of opposite sides are other. So, WY = XZ = 2c. All four angles of a congruent, so the sign is a parallelogram, but no rectangle are right angles. So, is a right measure is given that can be used to prove that it is a rectangle. triangle. By the Pythagorean Theorem, XZ2 = XY2 + 2 ZY . PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let Therefore, WY = 2(5) = 10. the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, ANSWER: b), and C(a, 0). You need to walk through the proof 10 step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a 41. SIGNS The sign below is in the foyer of Nyoko’s rectangle and you need to prove . Use the school. Based on the dimensions given, can Nyoko properties that you have learned about rectangles to be sure that the sign is a rectangle? Explain your walk through the proof. reasoning. Given: ABCD is a rectangle. Prove:

SOLUTION:

Both pairs of opposite sides are congruent, so the Proof: sign is a parallelogram, but no measure is given that Use the Distance Formula to find the lengths of the can be used to prove that it is a rectangle. If the diagonals. diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle.

ANSWER: The diagonals, have the same length, so No; sample answer: Both pairs of opposite sides are they are congruent. congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a ANSWER: rectangle. Given: ABCD is a rectangle. Prove: PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof Proof: step by step. Look over what you are given and what Use the Distance Formula to find you need to prove. Here, you are given ABCD is a and . rectangle and you need to prove . Use the have the same length, so they are properties that you have learned about rectangles to congruent. walk through the proof.

Given: ABCD is a rectangle. 43. If the diagonals of a parallelogram are congruent, then it is a rectangle. Prove: SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given Proof: ABCD and and you need to prove that Use the Distance Formula to find the lengths of the diagonals. ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and The diagonals, have the same length, so Prove: ABCD is a rectangle. they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove:

Proof:

AC = BD. So, Proof: Use the Distance Formula to find and . have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, Because A and B are different points, then it is a rectangle. a ≠ 0. Then b = 0. The slope of is undefined SOLUTION: and the slope of . Thus, . ∠DAB is Begin by positioning parallelogram ABCD on a a right angle and ABCD is a rectangle. coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c ANSWER: units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the Given: ABCD and proof step by step. Look over what you are given Prove: ABCD is a rectangle. and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Proof: Prove: ABCD is a rectangle.

But AC = BD. So,

Proof:

AC = BD. So, Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each Because A and B are different points, with all four sides congruent. Label one a ≠ 0. Then b = 0. The slope of is undefined parallelogram ABCD, one MNOP, and one WXYZ. and the slope of . Thus, . ∠DAB is Draw the two diagonals of each parallelogram and a right angle and ABCD is a rectangle. label the intersections R. b. TABULAR Use a protractor to measure the ANSWER: appropriate angles and complete the table below. Given: ABCD and Prove: ABCD is a rectangle. c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

Proof:

But AC = BD. So,

b. Use a protractor to measure each angle listed in the table. Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined c. Sample answer: Each of the angles listed in the and the slope of . Thus, . ∠DAB is table are right angles. These angles were each a right angle and ABCD is a rectangle.. formed by diagonals. The diagonals of a parallelogram with four congruent sides are 44. MULTIPLE REPRESENTATIONS In the perpendicular. problem, you will explore properties of other special parallelograms. ANSWER: a. GEOMETRIC Draw three parallelograms, each a. Sample answer: with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer: b.

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

b. Use a protractor to measure each angle listed in the table.

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a SOLUTION: parallelogram with four congruent sides are All four angles of a rectangle are right angles. So, perpendicular. ANSWER: a. Sample answer: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So,

The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular. ANSWER: x = 6, y = −10 45. CHALLENGE In rectangle ABCD, Find the 46. CCSS CRITIQUE Parker says that any two values of x and y. congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

SOLUTION: All four angles of a rectangle are right angles. So,

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right The diagonals of a rectangle are congruent and angle. bisect each other. So, is an isosceles triangle with So,

So, Tamika is correct. The sum of the angles of a triangle is 180. ANSWER: So, Tamika; Sample answer: When two congruent By the Vertical Angle Theorem, triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

REASONING ANSWER: 47. In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. x = 6, y = −10 How many rectangles are formed by the intersecting lines? 46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

Then the rectangles formed are ABXW, ACYW, So, Tamika is correct. BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines. ANSWER: Tamika; Sample answer: When two congruent ANSWER: triangles are arranged to form a quadrilateral, two of 6 the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of 48. OPEN ENDED Write the equations of four lines the angles in the congruent triangles has to be a right having intersections that form the vertices of a angle. rectangle. Verify your answer using coordinate geometry. 47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. SOLUTION: How many rectangles are formed by the intersecting Sample answer: lines? A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines Then the rectangles formed are ABXW, ACYW, are perpendicular and the measure of the angle they BCYX, BDZX, CDZY, and ADZW. Therefore, 6 form is 90. By Theorem 6.6, if a parallelogram has rectangles are formed by the intersecting lines. one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. ANSWER: 6 ANSWER: Sample answer: x = 0, x = 6, y = 0, 48. OPEN ENDED Write the equations of four lines y = 4; having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the The length of is 6 − 0 or 6 units and the length of rectangle. The lines y = 4 and x = 6 form the other 2 is 6 – 0 or 6 units. The slope of AB is 0 and the sides. slope of DC is 0. Since a pair of sides of the

quadrilateral is both parallel and congruent, by

x = 0, x = 6, y = 0, y = 4; Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: ANSWER: Sample answer: x = 0, x = 6, y = 0, Sample answer: All rectangles are parallelograms y = 4; because, by definition, both pairs of opposite sides are parallel. Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram FGHJ a rectangle?

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the A x = 3, y = 4 slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by B x = 4, y = 3 Theorem 6.12, the quadrilateral is a parallelogram. C x = 7, y = 8 D Since is horizontal and is vertical, the lines x = 8, y = 7 are perpendicular and the measure of the angle they SOLUTION: form is 90. By Theorem 6.6, if a parallelogram has The opposite sides of a parallelogram are congruent. one right angle, it has four right angles. Therefore, by So, FJ = GH. definition, the parallelogram is a rectangle. –3x + 5y = 11 FJ = GH

WRITING IN MATH 49. Explain why all rectangles If the diagonals of a parallelogram are congruent, are parallelograms, but all parallelograms are not then it is a rectangle. Since FGHJ is a parallelogram,

rectangles. the diagonals bisect each other. So, if it is a SOLUTION: rectangle, then will be an isosceles triangle Sample answer: A parallelogram is a quadrilateral in with That is, 3x + y = 13. which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs Add the two equations to eliminate the x-term and of opposite sides parallel. So rectangles are always solve for y. parallelograms. –3x + 5y = 11 3x + y = 13 Parallelograms with right angles are rectangles, so 6y = 24 Add to eliminate x-term. some parallelograms are rectangles, but others with y = 4 Divide each side by 6. non-right angles are not. Use the value of y to find the value of x. ANSWER: 3x + 4 = 13 Substitute. Sample answer: All rectangles are parallelograms 3x = 9 Subtract 4 from each side. because, by definition, both pairs of opposite sides x = 3 Divide each side by 3. are parallel. Parallelograms with right angles are rectangles, so some parallelograms are rectangles, So, x = 3, y = 4, the correct choice is A. but others with non-right angles are not. ANSWER: 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = A 13, what values of x and y make parallelogram FGHJ a rectangle? 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following equations could be used to find r, the shorter side of the playground? F 10r + r = 80 A x = 3, y = 4 G 4r + 10 = 80 B x = 4, y = 3 H r(r + 10) = 80 C x = 7, y = 8 J 2(r + 10) + 2r = 80 D x = 8, y = 7 SOLUTION: SOLUTION: The formula to find the perimeter of a rectangle is The opposite sides of a parallelogram are congruent. given by P = 2(l + w). The perimeter is given to be 8

So, FJ = GH. ft. The width is r and the length is r + 10. So, 2r + 2 –3x + 5y = 11 FJ = GH (r + 10) = 80.

Therefore, the correct choice is J. If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, ANSWER: the diagonals bisect each other. So, if it is a J rectangle, then will be an isosceles triangle with That is, 3x + y = 13. 52. SHORT RESPONSE What is the measure of ? Add the two equations to eliminate the x-term and solve for y. –3x + 5y = 11 3x + y = 13 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6.

Use the value of y to find the value of x. SOLUTION: 3x + 4 = 13 Substitute. The diagonals of a rectangle are congruent and 3x = 9 Subtract 4 from each side. bisect each other. So, is an isosceles triangle x = 3 Divide each side by 3. with So, So, x = 3, y = 4, the correct choice is A. By the Exterior Angle Theorem,

ANSWER: A ANSWER: 112 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 53. SAT/ACT If p is odd, which of the following must feet longer than the other. Which of the following also be odd? equations could be used to find r, the shorter side of A 2p the playground? B 2p + 2 F 10r + r = 80 G 4r + 10 = 80 C H r(r + 10) = 80 D J 2(r + 10) + 2r = 80 E p + 2 SOLUTION: SOLUTION: The formula to find the perimeter of a rectangle is Analyze each answer choice to determine which given by P = 2(l + w). The perimeter is given to be 8 expression is odd when p is odd. ft. The width is r and the length is r + 10. So, 2r + 2 Choice A 2p (r + 10) = 80. This is a multiple of 2 so it is even. Therefore, the correct choice is J. Choice B 2p + 2 ANSWER: This can be rewritten as 2(p + 1) which is also a J multiple of 2 so it is even.

52. SHORT RESPONSE What is the measure of Choice C ? If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. SOLUTION: The correct choice is E. The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle with E

So, ALGEBRA Find x and y so that the By the Exterior Angle Theorem, quadrilateral is a parallelogram.

ANSWER: 112

53. SAT/ACT If p is odd, which of the following must 54. also be odd? SOLUTION: A 2p The opposite sides of a parallelogram are congruent. B 2p + 2

D In a parallelogram, consecutive interior angles are E p + 2 supplementary. SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a ANSWER: multiple of 2 so it is even. x = 2, y = 41

Choice C If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 55. Adding 2 to an odd number results in an odd SOLUTION: number. This is odd. The opposite sides of a parallelogram are congruent. The correct choice is E. ANSWER: E The opposite sides of a parallelogram are parallel to ALGEBRA Find x and y so that the each other. So, the angles with the measures (3y + quadrilateral is a parallelogram. 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

54. ANSWER: SOLUTION: x = 8, y = 22 The opposite sides of a parallelogram are congruent.

In a parallelogram, consecutive interior angles are supplementary. 56.

SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 ANSWER: Use the value of y in the second equation and solve x = 2, y = 41 for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: x = 2, y = 7 55. 57. COORDINATE GEOMETRY Find the SOLUTION: coordinates of the intersection of the diagonals of The opposite sides of a parallelogram are congruent. parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: The diagonals of a parallelogram bisect each other, The opposite sides of a parallelogram are parallel to so the intersection is the midpoint of either diagonal. each other. So, the angles with the measures (3y + Find the midpoint of or . 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: ANSWER: (2.5, 0.5) x = 8, y = 22 Refer to the figure.

56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 58. If , name two congruent angles. 2y = 14 y = 7 SOLUTION: Use the value of y in the second equation and solve Since is an isosceles triangle. So, for x. 7 + 3 = 2x + 6 4 = 2x ANSWER: 2 = x ANSWER: x = 2, y = 7 59. If , name two congruent segments.

57. COORDINATE GEOMETRY Find the SOLUTION: coordinates of the intersection of the diagonals of In a triangle with two congruent angles, the sides parallelogram ABCD with vertices A(1, 3), B(6, 2), C opposite to the congruent angles are also congruent. (4, –2), and D(–1, –1). So, .

SOLUTION: ANSWER: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or . 60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides Therefore, the coordinates of the intersection of the opposite to the congruent angles are also congruent. diagonals are (2.5, 0.5). So, . ANSWER: ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: FARMING An X-brace on a rectangular barn The diagonals of a rectangle bisect each other. So, door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

ANSWER:

SOLUTION: 1. QR The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, is an isosceles triangle. The opposite sides of a rectangle are parallel and Then, congruent. Therefore, QR = PS = 7 ft. By the Exterior Angle Theorem,

ANSWER: 7 ft Therefore,

2. SQ ANSWER: 33.5 SOLUTION: The diagonals of a rectangle bisect each other. So, 4.

SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore, ANSWER: ANSWER:

56.5 ALGEBRA Quadrilateral DEFG is a rectangle. 3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, 5. If FD = 3x – 7 and EG = x + 5, find EG. By the Exterior Angle Theorem, SOLUTION: The diagonals of a rectangle are congruent to each Therefore, other. So, FD = EG.

ANSWER: 33.5 Use the value of x to find EG. 4. EG = 6 + 5 = 11 SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 11 bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem, 6. If , find .

Therefore, SOLUTION: All four angles of a rectangle are right angles. So, ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

ANSWER:

5. If FD = 3x – 7 and EG = x + 5, find EG. 51 SOLUTION: 7. PROOF If ABDE is a rectangle and , The diagonals of a rectangle are congruent to each prove that . other. So, FD = EG.

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 11 SOLUTION: You need to walk through the proof step by step. 6. If , Look over what you are given and what you need to prove. You are given ABDE is a rectangle and find . . You need to prove . Use the SOLUTION: properties that you have learned about rectangles to All four angles of a rectangle are right angles. So, walk through the proof.

Given: ABDE is a rectangle; Prove:

ANSWER: 51

7. PROOF If ABDE is a rectangle and , Statements(Reasons)

prove that . 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS)

SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. ANSWER: Look over what you are given and what you need to prove. You are given ABDE is a rectangle and Given: ABDE is a rectangle;

. You need to prove . Use the Prove: properties that you have learned about rectangles to walk through the proof.

Given: ABDE is a rectangle; Prove:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle)

Statements(Reasons) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 1. ABDE is a rectangle; . (Given) 5. (All rt .) 2. ABDE is a parallelogram. (Def. of 6. (SAS)

rectangle) 7. (CPCTC) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine 5. (All rt .) whether the figure is a rectangle. Justify your 6. (SAS) answer using the indicated formula. 7. (CPCTC) 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula

ANSWER: SOLUTION: Given: ABDE is a rectangle; Prove:

Statements(Reasons)

1. ABDE is a rectangle; . Use the slope formula to find the slope of the sides of (Given) the quadrilateral. 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine There are no parallel sides, so WXYZ is not a whether the figure is a rectangle. Justify your parallelogram. Therefore, WXYZ is not a rectangle. answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula ANSWER: SOLUTION: No; slope of , slope of , slope of

, and slope of . Slope of

slope of , and slope of slope of , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION: There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of

slope of , and slope of slope of , so WXYZ is not a parallelogram. Therefore,

WXYZ is not a rectangle. First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula So the diagonals are congruent. Thus, ABCD is a rectangle. SOLUTION: ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, so ABCD is a AD = 2 feet, and , find each parallelogram. measure. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a 10. BC parallelogram. , so the diagonals are SOLUTION: congruent. Thus, ABCD is a rectangle. The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: 2 ft

11. DB SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC .

FENCING X-braces are also used to provide BC = AD = 2 support in rectangular fencing. If AB = 6 feet, DC = AB = 6 AD = 2 feet, and , find each measure.

ANSWER: 6.3 ft

12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles 10. BC triangle. SOLUTION: Then,

The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: By the Vertical Angle Theorem, 2 ft ANSWER: 11. DB 50 SOLUTION: All four angles of a rectangle are right angles. So, 13. is a right triangle. By the Pythagorean SOLUTION: 2 2 2 Theorem, DB = BC + DC . The diagonals of a rectangle are congruent and BC = AD = 2 bisect each other. So, is an isosceles DC = AB = 6 triangle. Then, All four angles of a rectangle are right angles.

ANSWER: 6.3 ft ANSWER: 12. 25 SOLUTION: CCSS REGULARITY Quadrilateral WXYZ is a The diagonals of a rectangle are congruent and rectangle. bisect each other. So, is an isosceles triangle. Then,

By the Vertical Angle Theorem, 14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: ANSWER: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 50 2x + 3 = x + 4 13. x = 1 Use the value of x to find WX. SOLUTION: WX = 1 + 4 = 5. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles ANSWER: triangle. 5 Then, All four angles of a rectangle are right angles. 15. If PY = 3x – 5 and WP = 2x + 11, find ZP.

SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP. ANSWER: 25 The diagonals of a rectangle are congruent and CCSS REGULARITY Quadrilateral WXYZ is a bisect each other. So, is an isosceles triangle. rectangle.

ANSWER: 43

14. If ZY = 2x + 3 and WX = x + 4, find WX. 16. If , find SOLUTION: . The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. SOLUTION: 2x + 3 = x + 4 All four angles of a rectangle are right angles. So, x = 1 Use the value of x to find WX. WX = 1 + 4 = 5.

ANSWER:

15. If PY = 3x – 5 and WP = 2x + 11, find ZP. ANSWER: SOLUTION: 39 The diagonals of a rectangle bisect each other. So, PY = WP. 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: The diagonals of a rectangle are congruent and The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY. bisect each other. So, is an isosceles triangle.

Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38. ANSWER: 43 ANSWER: 38 16. If , find . 18. If , find . SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: ANSWER: 39 48 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. 19. If , find . SOLUTION: The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, ZP = PY. All four angles of a rectangle are right angles. So, is a right triangle. Then,

Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38.

ANSWER: Therefore, Since is a 38 transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, 18. If , find = = 46. . ANSWER: SOLUTION: 46 All four angles of a rectangle are right angles. So, PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. Prove:

ANSWER:

48 SOLUTION: 19. If , find You need to walk through the proof step by step. . Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. SOLUTION: You need to prove . Use the All four angles of a rectangle are right angles. So, properties that you have learned about rectangles to is a right triangle. Then, walk through the proof.

Given: ABCD is a rectangle. Prove: Proof: Statements (Reasons) Therefore, Since is a 1. ABCD is a rectangle. (Given) transversal of parallel sides and , alternate 2. ABCD is a parallelogram. (Def. of rectangle) interior angles WZX and ZXY are congruent. So, 3. (Opp. sides of a .) = = 46. 4. (Refl. Prop.)

ANSWER: 5. (Diagonals of a rectangle are .) 46 6. (SSS) PROOF Write a two-column proof. ANSWER: 20. Given: ABCD is a rectangle. Proof: Prove: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.)

5. (Diagonals of a rectangle are .) SOLUTION: 6. (SSS) You need to walk through the proof step by step. Look over what you are given and what you need to 21. Given: QTVW is a rectangle. prove. Here, you are given ABCD is a rectangle.

You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove: Prove: Proof: Statements (Reasons) SOLUTION: 1. ABCD is a rectangle. (Given) You need to walk through the proof step by step. 2. ABCD is a parallelogram. (Def. of rectangle) Look over what you are given and what you need to 3. (Opp. sides of a .) prove. Here, you are given QTVW is a rectangle and 4. (Refl. Prop.) . You need to prove . Use 5. (Diagonals of a rectangle are .) the properties that you have learned about rectangles 6. (SSS) to walk through the proof.

ANSWER: Given: QTVW is a rectangle. Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.) Prove:

prove. Here, you are given QTVW is a rectangle and 12. (Def. of segs.) . You need to prove . Use 13. (SAS) the properties that you have learned about rectangles ANSWER: to walk through the proof. Proof: Given: QTVW is a rectangle. Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .)

Prove: 6. QR = ST (Def. of segs.) Proof: 7. (Refl. Prop.) Statements (Reasons) 8. RS = RS (Def. of segs.) 1. QTVW is a rectangle; .(Given) 9. QR + RS = RS + ST (Add. prop.) 2. QTVW is a parallelogram. (Def. of rectangle) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 3. (Opp sides of a .) 12. (Def. of segs.) 4. are right angles.(Def. of rectangle) 13. (SAS) 5. (All rt .) 6. QR = ST (Def. of segs.) COORDINATE GEOMETRY Graph each 7. (Refl. Prop.) quadrilateral with the given vertices. Determine 8. RS = RS (Def. of segs.) whether the figure is a rectangle. Justify your 9. QR + RS = RS + ST (Add. prop.) answer using the indicated formula. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula 11. QS = RT (Subst.) SOLUTION: 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) Use the slope formula to find the slope of the sides of 6. QR = ST (Def. of segs.) the quadrilateral. 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine The slopes of each pair of opposite sides are equal. whether the figure is a rectangle. Justify your So, the two pairs of opposite sides are parallel. answer using the indicated formula. Therefore, the quadrilateral WXYZ is a parallelogram. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula The products of the slopes of the adjacent sides are SOLUTION: –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle.

ANSWER:

Yes; slope of slope of , slope

of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral.

The slopes of each pair of opposite sides are equal. 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance So, the two pairs of opposite sides are parallel. Formula Therefore, the quadrilateral WXYZ is a parallelogram. The products of the slopes of the adjacent sides are SOLUTION: –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle.

ANSWER:

Yes; slope of slope of , slope

of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula So the diagonals are not congruent. Thus, JKLM is SOLUTION: not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so JKLM is a 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula parallelogram. A parallelogram is a rectangle if the diagonals are SOLUTION: congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle. First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula So the diagonals are congruent. Thus, QRST is a rectangle. SOLUTION: ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula so QRST is a SOLUTION: parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are

congruent. QRST is a rectangle. First, use the slope formula to find the lengths of the sides of the quadrilateral.

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula Therefore, the quadrilateral GHJK is a SOLUTION: parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle.

ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

First, use the slope formula to find the lengths of the sides of the quadrilateral.

The slopes of each pair of opposite sides are equal. Quadrilateral ABCD is a rectangle. Find each So, the two pairs of opposite sides are parallel. measure if . Therefore, the quadrilateral GHJK is a parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle.

ANSWER: 26. No; slope of slope of and slope of SOLUTION: slope of . So, GHJK is a All four angles of a rectangle are right angles. So, parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

ANSWER: 50

27. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. Quadrilateral ABCD is a rectangle. Find each ANSWER: measure if . 40

28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. 26. Therefore, m∠3 = m∠2 = 40. SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 40

29. SOLUTION: ANSWER: All four angles of a rectangle are right angles. So,

28. 30. SOLUTION: SOLUTION: The measures of angles 2 and 3 are equal as they All four angles of a rectangle are right angles. So, are alternate interior angles. Therefore, m∠3 = m∠2 = 40.

ANSWER: The measures of angles 1 and 4 are equal as they 40 are alternate interior angles. Therefore, m 4 = m 1 = 50. ∠ ∠ 29. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 All four angles of a rectangle are right angles. So, and 6 is an isosceles triangle. Therefore, m∠6 = m∠4 = 50. ANSWER: 50 The measures of angles 1 and 4 are equal as they are alternate interior angles. 31. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 The measures of angles 2 and 3 are equal as they and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. are alternate interior angles. The sum of the three angles of a triangle is 180. Since the diagonals of a rectangle are congruent and Therefore, m∠5 = 180 – (50 + 50) = 80. bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. ANSWER: The sum of the three angles of a triangle is 180. 80 Therefore, m∠8 = 180 – (40 + 40) = 100.

30. ANSWER: 100 SOLUTION: All four angles of a rectangle are right angles. So, 32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one shown. To what length should she cut the metal supports in order for the bookshelf to be square, The measures of angles 1 and 4 are equal as they which means that the angles formed by the shelves are alternate interior angles. and the vertical supports are all right angles? Explain your reasoning. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = m∠4 = 50. ANSWER: 50

31. SOLUTION: SOLUTION: In order for the bookshelf to be square, the lengths of The measures of angles 2 and 3 are equal as they all of the metal supports should be equal. Since we are alternate interior angles. know the length of the bookshelves and the distance Since the diagonals of a rectangle are congruent and between them, use the Pythagorean Theorem to find bisect each other, the triangle with the angles 3, 7 the lengths of the metal supports. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. Let x be the length of each support. Express 3 feet The sum of the three angles of a triangle is 180. as 36 inches. Therefore, m∠8 = 180 – (40 + 40) = 100. ANSWER: 100

32. CCSS MODELING Jody is building a new Therefore, the metal supports should each be 39 bookshelf using wood and metal supports like the one inches long or 3 feet 3 inches long. shown. To what length should she cut the metal supports in order for the bookshelf to be square, ANSWER: which means that the angles formed by the shelves 3 ft 3 in.; Sample answer: In order for the bookshelf and the vertical supports are all right angles? Explain to be square, the lengths of all of the metal supports your reasoning. should be equal. Since I knew the length of the bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the metal supports.

PROOF Write a two-column proof. 33. Theorem 6.13 SOLUTION: You need to walk through the proof step by step. SOLUTION: Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we diagonals . You need to prove know the length of the bookshelves and the distance . Use the properties that you have learned between them, use the Pythagorean Theorem to find about rectangles to walk through the proof. the lengths of the metal supports. Let x be the length of each support. Express 3 feet Given: WXYZ is a rectangle with diagonals as 36 inches. . Prove:

Therefore, the metal supports should each be 39 inches long or 3 feet 3 inches long. Proof: ANSWER: 1. WXYZ is a rectangle with diagonals . 3 ft 3 in.; Sample answer: In order for the bookshelf (Given) to be square, the lengths of all of the metal supports 2. (Opp. sides of a .) should be equal. Since I knew the length of the bookshelves and the distance between them, I used 3. (Refl. Prop.) the Pythagorean Theorem to find the lengths of the 4. are right angles. (Def. of metal supports. rectangle)

PROOF Write a two-column proof. 5. (All right .) 33. Theorem 6.13 6. (SAS)

SOLUTION: 7. (CPCTC) You need to walk through the proof step by step. ANSWER: Look over what you are given and what you need to Given: WXYZ is a rectangle with diagonals prove. Here, you are given WXYZ is a rectangle with . diagonals . You need to prove

. Use the properties that you have learned Prove: about rectangles to walk through the proof.

Given: WXYZ is a rectangle with diagonals

. Prove: Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .)

3. (Refl. Prop.) Proof: 4. are right angles. (Def. of 1. WXYZ is a rectangle with diagonals . rectangle) (Given) 5. (All right .) 2. (Opp. sides of a .) 6. (SAS) 3. (Refl. Prop.) 7. (CPCTC) 4. are right angles. (Def. of rectangle) 34. Theorem 6.14 5. (All right .) SOLUTION:

6. (SAS) You need to walk through the proof step by step. 7. (CPCTC) Look over what you are given and what you need to prove. Here, you are given ANSWER: . You need to Given: WXYZ is a rectangle with diagonals prove that WXYZ is a rectangle.. Use the properties . that you have learned about rectangles to walk Prove: through the proof.

Given: Prove: WXYZ is a rectangle.

Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) Proof:

3. (Refl. Prop.) 1. (Given)

4. are right angles. (Def. of 2. (SSS) rectangle) 3. (CPCTC) 5. (All right .) 4. (Def. of ) 6. (SAS) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 7. (CPCTC) 6. are supplementary. (Cons. are suppl.) 34. Theorem 6.14 7. (Def of suppl.) SOLUTION: 8. are right angles. (If You need to walk through the proof step by step. 2 and suppl., each is a rt. .) Look over what you are given and what you need to 9. are right angles. (If a has 1 prove. Here, you are given rt. , it has 4 rt. .) . You need to 10. WXYZ is a rectangle. (Def. of rectangle) prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk ANSWER: through the proof. Given:

Prove: WXYZ is a rectangle. Given: Prove: WXYZ is a rectangle.

Proof: 1. (Given) Proof: 2. (SSS) 1. (Given) 3. (CPCTC) 2. (SSS) 4. (Def. of ) 3. (CPCTC) 5. WXYZ is a parallelogram. (If both pairs of opp.

4. (Def. of ) sides are , then quad. is .) 5. WXYZ is a parallelogram. (If both pairs of opp. 6. are supplementary. (Cons. sides are , then quad. is .) are suppl.) 6. are supplementary. (Cons. 7. (Def of suppl.) are suppl.) 8. are right angles. (If 7. (Def of suppl.) 2 and suppl., each is a rt. .) 8. are right angles. (If 9. are right angles. (If a has 1 2 and suppl., each is a rt. .) rt. , it has 4 rt. .) 9. are right angles. (If a has 1 10. WXYZ is a rectangle. (Def. of rectangle) rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) PROOF Write a paragraph proof of each statement. ANSWER: 35. If a parallelogram has one right angle, then it is a Given: rectangle. Prove: WXYZ is a rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Proof: Then use the properties of parallelograms to walk through the proof. 1. (Given)

2. (SSS) 3. (CPCTC) 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. ABCD is a parallelogram, and is a right angle. are suppl.) Since ABCD is a parallelogram and has one right 7. (Def of suppl.) angle, then it has four right angles. So by the 8. are right angles. (If definition of a rectangle, ABCD is a rectangle. 2 and suppl., each is a rt. .) ANSWER: 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right SOLUTION: angle, then it has four right angles. So by the You need to walk through the proof step by step. definition of a rectangle, ABCD is a rectangle. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one 36. If a quadrilateral has four right angles, then it is a right angle. You need to prove that it is a rectangle. rectangle. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk SOLUTION: through the proof. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and rectangles to walk through the proof.

ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle.

ANSWER: ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

ANSWER: ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle.

36. If a quadrilateral has four right angles, then it is a ABCD is a quadrilateral with four right angles. rectangle. ABCD is a parallelogram because both pairs of SOLUTION: opposite angles are congruent. By definition of a You need to walk through the proof step by step. rectangle, ABCD is a rectangle. Look over what you are given and what you need to prove. Here, you are given a quadrilateral with four 37. CONSTRUCTION Construct a rectangle using the right angles. You need to prove that it is a rectangle. construction for congruent segments and the Let ABCD be a quadrilateral with four right angles. construction for a line perpendicular to another line Then use the properties of parallelograms and through a point on the line. Justify each step of the rectangles to walk through the proof. construction. SOLUTION: Sample answer: Step 1: Graph a line and place points P and Q on the line.

Step 2: Place the compass on point P. Using the

same compass setting, draw an arc to the left and ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of right of P that intersects . opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. ANSWER: Place the compass on each point of intersection and draw arcs that intersect above .

Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3.

ABCD is a quadrilateral with four right angles. Step 5: Repeat steps 2 and 3 using point Q. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a Step 6: Draw line b through point Q and the rectangle, ABCD is a rectangle. intersection of the arcs drawn in step 5.

37. CONSTRUCTION Construct a rectangle using the Step 7: Using any compass setting, put the compass construction for congruent segments and the at P and draw an arc above it that intersects line a. construction for a line perpendicular to another line Label the point of intersection R. through a point on the line. Justify each step of the construction. Step 8: Using the same compass setting, put the SOLUTION: compass at Q and draw an arc above it that Sample answer: intersects line b. Label the point of intersection S. Step 1: Graph a line and place points P and Q on the line. Step 9: Draw .

Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and right of P that intersects .

Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Place the compass on each point of intersection and draw arcs that intersect above .

Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3. Since and , the measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 5: Repeat steps 2 and 3 using point Q. . The same compass setting was used to Step 6: Draw line b through point Q and the locate points R and S, so . If one pair of intersection of the arcs drawn in step 5. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 7: Using any compass setting, put the compass A parallelogram with right angles is a rectangle. at P and draw an arc above it that intersects line a. Thus, PRSQ is a rectangle. Label the point of intersection R.

Step 8: Using the same compass setting, put the ANSWER: compass at Q and draw an arc above it that intersects line b. Label the point of intersection S. Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that Step 9: Draw . are perpendicular to the same line are parallel, so . The same compass setting was used to locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

Since and , the measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so . The same compass setting was used to locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. 38. SPORTS The end zone of a football field is 160 feet A parallelogram with right angles is a rectangle. wide and 30 feet long. Kyle is responsible for

Thus, PRSQ is a rectangle. painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a ANSWER: rectangle using only a tape measure.

SOLUTION: Sample answer: Since and , the Sample answer: If the diagonals of a parallelogram measure of angles P and Q is 90 degrees. Lines that are congruent, then it is a rectangle. He should are perpendicular to the same line are parallel, so measure the diagonals of the end zone and both . The same compass setting was used to pairs of opposite sides. If both pairs of opposite sides locate points R and S, so . If one pair of are congruent, then the quadrilateral is a opposite sides of a quadrilateral is both parallel and parallelogram. If the diagonals are congruent, then

congruent, then the quadrilateral is a parallelogram. the end zone is a rectangle. A parallelogram with right angles is a rectangle. ANSWER: Thus, PRSQ is a rectangle. Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: 38. SPORTS The end zone of a football field is 160 feet The diagonals of a rectangle are congruent to each wide and 30 feet long. Kyle is responsible for other. So, YW = XZ = b. All four angles of a painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is rectangle are right angles. So, is a right 2 2 the regulation size and be sure that it is also a triangle. By the Pythagorean Theorem, XZ = XW + 2 rectangle using only a tape measure. WZ . SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should Therefore, YW = 5. measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides ANSWER: are congruent, then the quadrilateral is a 5 parallelogram. If the diagonals are congruent, then the end zone is a rectangle. 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. ANSWER: SOLUTION: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the The diagonals of a rectangle are congruent to each diagonals and both pairs of opposite sides are other. So, WY = XZ = 2c. All four angles of a congruent, then the end zone is a rectangle. rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XY2 + ALGEBRA Quadrilateral WXYZ is a rectangle. 2 ZY .

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: Therefore, WY = 2(5) = 10. The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a ANSWER: rectangle are right angles. So, is a right 10 2 2 triangle. By the Pythagorean Theorem, XZ = XW + SIGNS 2 41. The sign below is in the foyer of Nyoko’s WZ . school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your reasoning.

Therefore, YW = 5.

ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY.

SOLUTION: SOLUTION: The diagonals of a rectangle are congruent to each other. So, WY = XZ = 2c. All four angles of a rectangle are right angles. So, is a right Both pairs of opposite sides are congruent, so the 2 2 sign is a parallelogram, but no measure is given that triangle. By the Pythagorean Theorem, XZ = XY + can be used to prove that it is a rectangle. If the 2 ZY . diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no Therefore, WY = 2(5) = 10. measure is given that can be used to prove that it is a rectangle. 6-4 RectanglesANSWER: 10 PROOF Write a coordinate proof of each statement. 41. SIGNS The sign below is in the foyer of Nyoko’s 42. The diagonals of a rectangle are congruent. school. Based on the dimensions given, can Nyoko SOLUTION: be sure that the sign is a rectangle? Explain your reasoning. Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to walk through the proof. SOLUTION:

Given: ABCD is a rectangle. Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that Prove: can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no

measure is given that can be used to prove that it is a Proof: rectangle. Use the Distance Formula to find the lengths of the diagonals. PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent.

SOLUTION: The diagonals, have the same length, so Begin by positioning quadrilateral ABCD on a they are congruent. coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b ANSWER: units. Then the rest of the vertices are A(0, b), B(a, Given: ABCD is a rectangle. b), and C(a, 0). You need to walk through the proof Prove: step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove: Proof: Use the Distance Formula to find and . have the same length, so they are congruent. eSolutions Manual - Powered by Cognero 43. If the diagonals of a parallelogram are congruent,Page 15 Proof: then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + The diagonals, have the same length, so a, c), and D(b, c). You need to walk through the they are congruent. proof step by step. Look over what you are given and what you need to prove. Here, you are given ANSWER: ABCD and and you need to prove that Given: ABCD is a rectangle. ABCD is a rectangle. Use the properties that you Prove: have learned about rectangles to walk through the proof.

Given: ABCD and Prove: ABCD is a rectangle.

Proof: Use the Distance Formula to find and . Proof: have the same length, so they are

congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. AC = BD. So, SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given Because A and B are different points, ABCD and and you need to prove that a ≠ 0. Then b = 0. The slope of is undefined ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the and the slope of . Thus, . ∠DAB is proof. a right angle and ABCD is a rectangle.

ANSWER: Given: ABCD and Given: ABCD and Prove: ABCD is a rectangle. Prove: ABCD is a rectangle.

Proof: Proof:

AC = BD. So, But AC = BD. So,

Because A and B are different points, Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle. a right angle and ABCD is a rectangle.. ANSWER: 44. MULTIPLE REPRESENTATIONS In the

Given: ABCD and problem, you will explore properties of other special Prove: ABCD is a rectangle. parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below. Proof:

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. But AC = BD. So, SOLUTION:

a. Sample answer:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the

problem, you will explore properties of other special b. Use a protractor to measure each angle listed in

parallelograms. the table. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and c. Sample answer: Each of the angles listed in the label the intersections R. table are right angles. These angles were each b. TABULAR Use a protractor to measure the formed by diagonals. The diagonals of a appropriate angles and complete the table below. parallelogram with four congruent sides are perpendicular.

ANSWER: c. VERBAL Make a conjecture about the diagonals a. Sample answer: of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

b. Use a protractor to measure each angle listed in c. Sample answer: The diagonals of a parallelogram the table. with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the c. Sample answer: Each of the angles listed in the values of x and y. table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: a. Sample answer:

SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So, b.

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular. The sum of the angles of a triangle is 180. 45. CHALLENGE In rectangle ABCD, So, Find the By the Vertical Angle Theorem, values of x and y.

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a SOLUTION: rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is All four angles of a rectangle are right angles. So, either of them correct? Explain your reasoning.

SOLUTION: When two congruent triangles are arranged to form a The diagonals of a rectangle are congruent and quadrilateral, two of the angles are formed by a bisect each other. So, is an isosceles triangle single vertex of a triangle. with So,

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right

The sum of the angles of a triangle is 180. angle. So, By the Vertical Angle Theorem,

So, Tamika is correct. ANSWER: x = 6, y = −10 ANSWER: Tamika; Sample answer: When two congruent 46. CCSS CRITIQUE Parker says that any two triangles are arranged to form a quadrilateral, two of congruent acute triangles can be arranged to make a the angles are formed by a single vertex of a triangle. rectangle. Tamika says that only two congruent right In order for the quadrilateral to be a rectangle, one of triangles can be arranged to make a rectangle. Is the angles in the congruent triangles has to be a right either of them correct? Explain your reasoning. angle. SOLUTION: 47. REASONING In the diagram at the right, lines n, p, When two congruent triangles are arranged to form a q,and r are parallel and lines l and m are parallel. quadrilateral, two of the angles are formed by a How many rectangles are formed by the intersecting single vertex of a triangle. lines?

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle. SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

So, Tamika is correct.

ANSWER: Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle. Then the rectangles formed are ABXW, ACYW, 47. REASONING In the diagram at the right, lines n, p, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 q,and r are parallel and lines l and m are parallel. rectangles are formed by the intersecting lines. How many rectangles are formed by the intersecting lines? ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION:

Sample answer: SOLUTION: A rectangle has 4 right angles and each pair of Let the lines n, p, q, and r intersect the line l at the opposite sides is parallel and congruent. Lines that points A, B, C, and D respectively. Similarly, let the run parallel to the x-axis will be parallel to each other lines intersect the line m at the points W, X, Y, and Z. and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines. The length of is 6 − 0 or 6 units and the length of ANSWER: is 6 – 0 or 6 units. The slope of AB is 0 and the 6 slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by 48. OPEN ENDED Write the equations of four lines Theorem 6.12, the quadrilateral is a parallelogram. having intersections that form the vertices of a rectangle. Verify your answer using coordinate Since is horizontal and is vertical, the lines geometry. are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has SOLUTION: one right angle, it has four right angles. Therefore, by Sample answer: definition, the parallelogram is a rectangle. A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that ANSWER: run parallel to the x-axis will be parallel to each other Sample answer: x = 0, x = 6, y = 0, and perpendicular to lines parallel to the y-axis. y = 4; Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

x = 0, x = 6, y = 0, y = 4;

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. The length of is 6 − 0 or 6 units and the length of Since is horizontal and is vertical, the lines is 6 – 0 or 6 units. The slope of AB is 0 and the are perpendicular and the measure of the angle they slope of DC is 0. Since a pair of sides of the form is 90. By Theorem 6.6, if a parallelogram has quadrilateral is both parallel and congruent, by one right angle, it has four right angles. Therefore, by Theorem 6.12, the quadrilateral is a parallelogram. definition, the parallelogram is a rectangle. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they 49. WRITING IN MATH Explain why all rectangles form is 90. By Theorem 6.6, if a parallelogram has are parallelograms, but all parallelograms are not one right angle, it has four right angles. Therefore, by rectangles. definition, the parallelogram is a rectangle. SOLUTION: ANSWER: Sample answer: A parallelogram is a quadrilateral in Sample answer: x = 0, x = 6, y = 0, which each pair of opposite sides are parallel. By y = 4; definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms.

Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

ANSWER: Sample answer: All rectangles are parallelograms because, by definition, both pairs of opposite sides The length of is 6 − 0 or 6 units and the length of are parallel. Parallelograms with right angles are rectangles, so some parallelograms are rectangles, is 6 – 0 or 6 units. The slope of AB is 0 and the but others with non-right angles are not. slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = Theorem 6.12, the quadrilateral is a parallelogram. 13, what values of x and y make parallelogram Since is horizontal and is vertical, the lines FGHJ a rectangle? are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

49. WRITING IN MATH Explain why all rectangles A x = 3, y = 4 are parallelograms, but all parallelograms are not B x = 4, y = 3 rectangles. C x = 7, y = 8 SOLUTION: D x = 8, y = 7 Sample answer: A parallelogram is a quadrilateral in SOLUTION: which each pair of opposite sides are parallel. By The opposite sides of a parallelogram are congruent. definition of rectangle, all rectangles have both pairs So, FJ = GH. of opposite sides parallel. So rectangles are always –3x + 5y = 11 FJ = GH

parallelograms.

If the diagonals of a parallelogram are congruent, Parallelograms with right angles are rectangles, so then it is a rectangle. Since FGHJ is a parallelogram, some parallelograms are rectangles, but others with the diagonals bisect each other. So, if it is a non-right angles are not. rectangle, then will be an isosceles triangle ANSWER: with That is, 3x + y = 13. Sample answer: All rectangles are parallelograms because, by definition, both pairs of opposite sides Add the two equations to eliminate the x-term and are parallel. Parallelograms with right angles are solve for y. rectangles, so some parallelograms are rectangles, –3x + 5y = 11 but others with non-right angles are not. 3x + y = 13 6y = 24 Add to eliminate x-term. 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = y = 4 Divide each side by 6. 13, what values of x and y make parallelogram FGHJ a rectangle? Use the value of y to find the value of x. 3x + 4 = 13 Substitute. 3x = 9 Subtract 4 from each side. x = 3 Divide each side by 3.

So, x = 3, y = 4, the correct choice is A. A x = 3, y = 4 B x = 4, y = 3 ANSWER: C x = 7, y = 8 A D x = 8, y = 7 51. ALGEBRA A rectangular playground is surrounded SOLUTION: by an 80-foot fence. One side of the playground is 10 The opposite sides of a parallelogram are congruent. feet longer than the other. Which of the following So, FJ = GH. equations could be used to find r, the shorter side of –3x + 5y = 11 FJ = GH the playground? F 10r + r = 80 If the diagonals of a parallelogram are congruent, G 4r + 10 = 80 then it is a rectangle. Since FGHJ is a parallelogram, H r(r + 10) = 80 the diagonals bisect each other. So, if it is a J 2(r + 10) + 2r = 80 rectangle, then will be an isosceles triangle with That is, 3x + y = 13. SOLUTION: The formula to find the perimeter of a rectangle is Add the two equations to eliminate the x-term and given by P = 2(l + w). The perimeter is given to be 8 solve for y. ft. The width is r and the length is r + 10. So, 2r + 2 –3x + 5y = 11 (r + 10) = 80. 3x + y = 13 Therefore, the correct choice is J. 6y = 24 Add to eliminate x-term. ANSWER: y = 4 Divide each side by 6. J Use the value of y to find the value of x. SHORT RESPONSE 3x + 4 = 13 Substitute. 52. What is the measure of 3x = 9 Subtract 4 from each side. ? x = 3 Divide each side by 3.

So, x = 3, y = 4, the correct choice is A.

ANSWER: A SOLUTION: 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 The diagonals of a rectangle are congruent and feet longer than the other. Which of the following bisect each other. So, is an isosceles triangle equations could be used to find r, the shorter side of with the playground? So, F 10r + r = 80 By the Exterior Angle Theorem, G 4r + 10 = 80 H r(r + 10) = 80 ANSWER: J 2(r + 10) + 2r = 80 112 SOLUTION: The formula to find the perimeter of a rectangle is 53. SAT/ACT If p is odd, which of the following must given by P = 2(l + w). The perimeter is given to be 8 also be odd? ft. The width is r and the length is r + 10. So, 2r + 2 A 2p (r + 10) = 80. B 2p + 2 Therefore, the correct choice is J. C ANSWER: D J E p + 2 52. SHORT RESPONSE What is the measure of SOLUTION: ? Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even. SOLUTION: Choice C The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle If p is odd , is not a whole number. with Choice D 2p – 2 So, This can be rewritten as 2(p - 1) which is also a By the Exterior Angle Theorem, multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd ANSWER: number. This is odd. 112 The correct choice is E.

53. SAT/ACT If p is odd, which of the following must ANSWER: also be odd? E A 2p B 2p + 2 ALGEBRA Find x and y so that the quadrilateral is a parallelogram. C D E p + 2

SOLUTION: 54. Analyze each answer choice to determine which expression is odd when p is odd. SOLUTION: Choice A 2p The opposite sides of a parallelogram are congruent. This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even. In a parallelogram, consecutive interior angles are supplementary. Choice C If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd ANSWER: number. This is odd. x = 2, y = 41 The correct choice is E.

ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram. 55. SOLUTION: The opposite sides of a parallelogram are congruent.

54.

SOLUTION: The opposite sides of a parallelogram are parallel to The opposite sides of a parallelogram are congruent. each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

In a parallelogram, consecutive interior angles are supplementary.

ANSWER: x = 8, y = 22

ANSWER: x = 2, y = 41 56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14

55. y = 7 SOLUTION: Use the value of y in the second equation and solve The opposite sides of a parallelogram are congruent. for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

The opposite sides of a parallelogram are parallel to ANSWER: each other. So, the angles with the measures (3y + x = 2, y = 7 3)° and (4y – 19)° are alternate interior angles and hence they are equal. 57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: ANSWER: The diagonals of a parallelogram bisect each other, x = 8, y = 22 so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the 56. diagonals are (2.5, 0.5). SOLUTION: ANSWER: The diagonals of a parallelogram bisect each other. (2.5, 0.5) So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. Refer to the figure. 2y = 14 y = 7 Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: x = 2, y = 7 58. If , name two congruent angles. SOLUTION: 57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of Since is an isosceles triangle. So, parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). ANSWER: SOLUTION:

The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or . 59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides

Therefore, the coordinates of the intersection of the opposite to the congruent angles are also congruent. diagonals are (2.5, 0.5). So, .

ANSWER: ANSWER: (2.5, 0.5) Refer to the figure. 60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps Then, to prevent the door from warping over time. By the Exterior Angle Theorem,

If feet, PS = 7 feet, and

Therefore, , find each measure.

ANSWER: 33.5

4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

1. QR Therefore, SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft. ANSWER: 56.5 ANSWER: 7 ft ALGEBRA Quadrilateral DEFG is a rectangle.

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So, 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. ANSWER: EG = 6 + 5 = 11

ANSWER: 11 3. 6. If , SOLUTION: find . The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. SOLUTION: Then, All four angles of a rectangle are right angles. So, By the Exterior Angle Theorem,

Therefore,

ANSWER: 33.5 ANSWER: 4. 51 SOLUTION: The diagonals of a rectangle are congruent and 7. PROOF If ABDE is a rectangle and , bisect each other. So, is an isosceles triangle. prove that . Then, By the Vertical Angle Theorem,

Therefore,

ANSWER: 56.5 SOLUTION: ALGEBRA Quadrilateral DEFG is a rectangle. You need to walk through the proof step by step. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and . You need to prove . Use the properties that you have learned about rectangles to walk through the proof. 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: Given: ABDE is a rectangle; The diagonals of a rectangle are congruent to each Prove: other. So, FD = EG.

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: Statements(Reasons) 11 1. ABDE is a rectangle; . (Given) 6. If , 2. ABDE is a parallelogram. (Def. of rectangle) find . 3. (Opp. sides of a .) SOLUTION: 4. are right angles.(Def. of rectangle) All four angles of a rectangle are right angles. So, 5. (All rt .)

6. (SAS)

7. (CPCTC)

ANSWER:

Given: ABDE is a rectangle; Prove: ANSWER: 51

7. PROOF If ABDE is a rectangle and , prove that .

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) SOLUTION: 4. are right angles.(Def. of rectangle) You need to walk through the proof step by step. 5. (All rt .) Look over what you are given and what you need to 6. (SAS) prove. You are given ABDE is a rectangle and 7. (CPCTC) . You need to prove . Use the properties that you have learned about rectangles to COORDINATE GEOMETRY Graph each walk through the proof. quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. Given: ABDE is a rectangle; 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula Prove: SOLUTION:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) Use the slope formula to find the slope of the sides of 4. are right angles.(Def. of rectangle) the quadrilateral.

5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of

Statements(Reasons) , and slope of . Slope of 1. ABDE is a rectangle; .

(Given) slope of , and slope of slope of 2. ABDE is a parallelogram. (Def. of rectangle) , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION: 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so ABCD is a There are no parallel sides, so WXYZ is not a parallelogram. parallelogram. Therefore, WXYZ is not a rectangle. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the ANSWER: lengths of the diagonals.

No; slope of , slope of , slope of So the diagonals are congruent. Thus, ABCD is a , and slope of . Slope of rectangle.

slope of , and slope of slope of ANSWER: , so WXYZ is not a parallelogram. Therefore, Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a WXYZ is not a rectangle. parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, SOLUTION: AD = 2 feet, and , find each measure.

10. BC First, Use the Distance formula to find the lengths of the sides of the quadrilateral. SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: 2 ft so ABCD is a parallelogram. 11. DB A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the SOLUTION: lengths of the diagonals. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . So the diagonals are congruent. Thus, ABCD is a BC = AD = 2 rectangle. DC = AB = 6

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. ANSWER: 6.3 ft

12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

FENCING X-braces are also used to provide By the Vertical Angle Theorem, support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each ANSWER: measure. 50

13. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, All four angles of a rectangle are right angles. 10. BC SOLUTION: The opposite sides of a rectangle are parallel and

congruent. Therefore, BC = AD = 2 ft. ANSWER: ANSWER: 25 2 ft CCSS REGULARITY Quadrilateral WXYZ is a rectangle. 11. DB SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 14. If ZY = 2x + 3 and WX = x + 4, find WX. DC = AB = 6 SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 ANSWER: Use the value of x to find WX. 6.3 ft WX = 1 + 4 = 5.

12. ANSWER: SOLUTION: 5 The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. triangle. SOLUTION: Then, The diagonals of a rectangle bisect each other. So, PY = WP.

By the Vertical Angle Theorem, The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. ANSWER: 50

13. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 43 bisect each other. So, is an isosceles 16. If , find triangle. . Then, All four angles of a rectangle are right angles. SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 25

CCSS REGULARITY Quadrilateral WXYZ is a rectangle. ANSWER: 39 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX.

SOLUTION: 14. If ZY = 2x + 3 and WX = x + 4, find WX. The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY. SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 2x + 3 = x + 4

x = 1 Then ZP = 4(7) – 9 = 19.

Use the value of x to find WX. Therefore, ZX = 2(ZP) = 38. WX = 1 + 4 = 5. ANSWER: ANSWER: 38 5 18. If , find . 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, The diagonals of a rectangle bisect each other. So, PY = WP.

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

ANSWER: 48

19. If , find ANSWER: . 43 SOLUTION: 16. If , find All four angles of a rectangle are right angles. So, . is a right triangle. Then, SOLUTION: All four angles of a rectangle are right angles. So,

Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: ANSWER: 39 46 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. SOLUTION: Prove: The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY.

Then ZP = 4(7) – 9 = 19. SOLUTION:

Therefore, ZX = 2(ZP) = 38. You need to walk through the proof step by step. ANSWER: Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. 38 You need to prove . Use the 18. If , find properties that you have learned about rectangles to walk through the proof.

. SOLUTION: Given: ABCD is a rectangle. All four angles of a rectangle are right angles. So, Prove: Proof:

Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .)

4. (Refl. Prop.) ANSWER: 5. (Diagonals of a rectangle are .) 48 6. (SSS)

19. If , find ANSWER: . Proof: Statements (Reasons) SOLUTION: 1. ABCD is a rectangle. (Given) All four angles of a rectangle are right angles. So, 2. ABCD is a parallelogram. (Def. of rectangle) is a right triangle. Then, 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS)

Therefore, Since is a 21. Given: QTVW is a rectangle. transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, = = 46.

ANSWER: 46 Prove: PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. SOLUTION: Prove: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and . You need to prove . Use the properties that you have learned about rectangles to walk through the proof. SOLUTION: You need to walk through the proof step by step. Given: QTVW is a rectangle. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove: Prove: Proof: Proof: Statements (Reasons) Statements (Reasons) 1. QTVW is a rectangle; .(Given) 1. ABCD is a rectangle. (Given) 2. QTVW is a parallelogram. (Def. of rectangle) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle)

4. (Refl. Prop.) 5. (All rt .)

5. (Diagonals of a rectangle are .) 6. QR = ST (Def. of segs.) 6. (SSS) 7. (Refl. Prop.) ANSWER: 8. RS = RS (Def. of segs.) Proof: 9. QR + RS = RS + ST (Add. prop.) Statements (Reasons) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 1. ABCD is a rectangle. (Given) 11. QS = RT (Subst.) 2. ABCD is a parallelogram. (Def. of rectangle) 12. (Def. of segs.) 3. (Opp. sides of a .) 13. (SAS) 4. (Refl. Prop.) ANSWER:

5. (Diagonals of a rectangle are .) Proof: 6. (SSS) Statements (Reasons) 1. QTVW is a rectangle; .(Given) 21. Given: QTVW is a rectangle. 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) Prove: 8. RS = RS (Def. of segs.) SOLUTION: 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) You need to walk through the proof step by step. 11. QS = RT (Subst.) Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and 12. (Def. of segs.) . You need to prove . Use 13. (SAS) the properties that you have learned about rectangles to walk through the proof. COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine Given: QTVW is a rectangle. whether the figure is a rectangle. Justify your answer using the indicated formula.

22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION:

Prove: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) Use the slope formula to find the slope of the sides of 6. QR = ST (Def. of segs.) the quadrilateral. 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: The slopes of each pair of opposite sides are equal. Statements (Reasons) So, the two pairs of opposite sides are parallel. 1. QTVW is a rectangle; .(Given) Therefore, the quadrilateral WXYZ is a parallelogram. 2. QTVW is a parallelogram. (Def. of rectangle) The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to 3. (Opp sides of a .) each other. That is, all the four angles are right 4. are right angles.(Def. of rectangle) angles. Therefore, WXYZ is a rectangle. 5. (All rt .) ANSWER: 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) Yes; slope of slope of , slope 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) of slope of . So WXYZ is a 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) parallelogram. The product of the slopes of 11. QS = RT (Subst.) consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a 12. (Def. of segs.) rectangle. 13. (SAS)

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to so JKLM is a each other. That is, all the four angles are right parallelogram. angles. Therefore, WXYZ is a rectangle. A parallelogram is a rectangle if the diagonals are ANSWER: congruent. Use the Distance formula to find the lengths of the diagonals. Yes; slope of slope of , slope

of slope of . So WXYZ is a parallelogram. The product of the slopes of So the diagonals are not congruent. Thus, JKLM is

consecutive sides is –1, so the consecutive sides are not a rectangle. perpendicular and form right angles. Thus, WXYZ is a ANSWER: rectangle. No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula

SOLUTION: 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are so QRST is a congruent. Use the Distance formula to find the

lengths of the diagonals. parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals. So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: So the diagonals are congruent. Thus, QRST is a rectangle. No; so JKLM is a parallelogram; so ANSWER: the diagonals are not congruent. Thus, JKLM is not a Yes; so QRST is a rectangle. parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION: SOLUTION:

First, use the Distance Formula to find the lengths of

the sides of the quadrilateral. First, use the slope formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

The slopes of each pair of opposite sides are equal. So the diagonals are congruent. Thus, QRST is a So, the two pairs of opposite sides are parallel. rectangle. Therefore, the quadrilateral GHJK is a

ANSWER: parallelogram. None of the adjacent sides have slopes whose Yes; so QRST is a product is –1. So, the angles are not right angles. parallelogram. so the diagonals are Therefore, GHJK is not a rectangle. congruent. QRST is a rectangle. ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

First, use the slope formula to find the lengths of the sides of the quadrilateral. 26. SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 50

The slopes of each pair of opposite sides are equal.

So, the two pairs of opposite sides are parallel. 27. Therefore, the quadrilateral GHJK is a SOLUTION: parallelogram. The measures of angles 2 and 3 are equal as they None of the adjacent sides have slopes whose are alternate interior angles. product is –1. So, the angles are not right angles. Since the diagonals of a rectangle are congruent and Therefore, GHJK is not a rectangle. bisect each other, the triangle with the angles 3, 7 ANSWER: and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. No; slope of slope of and slope of ANSWER: slope of . So, GHJK is a 40 parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are 28. not perpendicular. Thus, GHJK is not a rectangle. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

Quadrilateral ABCD is a rectangle. Find each measure if . The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. 26. Therefore, m∠5 = 180 – (50 + 50) = 80. SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 80

30. SOLUTION: All four angles of a rectangle are right angles. So, ANSWER:

27. The measures of angles 1 and 4 are equal as they SOLUTION: are alternate interior angles. The measures of angles 2 and 3 are equal as they Therefore, m∠4 = m∠1 = 50. are alternate interior angles. Since the diagonals of a rectangle are congruent and Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 bisect each other, the triangle with the angles 3, 7 and 6 is an isosceles triangle. Therefore, m∠6 = and 8 is an isosceles triangle. So, m∠7 = m∠3. m∠4 = 50. Therefore, m 7 = m 2 = 40. ∠ ∠ ANSWER: ANSWER: 50 40 31. 28. SOLUTION: SOLUTION: The measures of angles 2 and 3 are equal as they The measures of angles 2 and 3 are equal as they are alternate interior angles. are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 Therefore, m∠3 = m∠2 = 40. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. ANSWER: The sum of the three angles of a triangle is 180. 40 Therefore, m∠8 = 180 – (40 + 40) = 100.

29. ANSWER: 100 SOLUTION: All four angles of a rectangle are right angles. So, 32. CCSS MODELING Jody is building a new bookshelf using wood and metal supports like the one shown. To what length should she cut the metal supports in order for the bookshelf to be square, which means that the angles formed by the shelves The measures of angles 1 and 4 are equal as they and the vertical supports are all right angles? Explain are alternate interior angles. your reasoning. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: 80 SOLUTION: 30. In order for the bookshelf to be square, the lengths of SOLUTION: all of the metal supports should be equal. Since we All four angles of a rectangle are right angles. So, know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find

the lengths of the metal supports. Let x be the length of each support. Express 3 feet as 36 inches. The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m 6 = ∠ Therefore, the metal supports should each be 39 m∠4 = 50. inches long or 3 feet 3 inches long.

ANSWER: ANSWER: 50 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports

31. should be equal. Since I knew the length of the SOLUTION: bookshelves and the distance between them, I used The measures of angles 2 and 3 are equal as they the Pythagorean Theorem to find the lengths of the are alternate interior angles. metal supports. Since the diagonals of a rectangle are congruent and PROOF Write a two-column proof. bisect each other, the triangle with the angles 3, 7 33. Theorem 6.13 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. SOLUTION: Therefore, m∠8 = 180 – (40 + 40) = 100. You need to walk through the proof step by step. Look over what you are given and what you need to ANSWER: prove. Here, you are given WXYZ is a rectangle with 100 diagonals . You need to prove 32. CCSS MODELING Jody is building a new . Use the properties that you have learned bookshelf using wood and metal supports like the one about rectangles to walk through the proof. shown. To what length should she cut the metal supports in order for the bookshelf to be square, Given: WXYZ is a rectangle with diagonals which means that the angles formed by the shelves . and the vertical supports are all right angles? Explain your reasoning. Prove:

Proof: 1. WXYZ is a rectangle with diagonals .

(Given) 2. (Opp. sides of a .) SOLUTION:

In order for the bookshelf to be square, the lengths of 3. (Refl. Prop.) all of the metal supports should be equal. Since we 4. are right angles. (Def. of know the length of the bookshelves and the distance rectangle) between them, use the Pythagorean Theorem to find 5. (All right .)

the lengths of the metal supports. 6. (SAS) Let x be the length of each support. Express 3 feet as 36 inches. 7. (CPCTC)

ANSWER: Given: WXYZ is a rectangle with diagonals . Prove: Therefore, the metal supports should each be 39 inches long or 3 feet 3 inches long.

ANSWER: 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports Proof: should be equal. Since I knew the length of the bookshelves and the distance between them, I used 1. WXYZ is a rectangle with diagonals . the Pythagorean Theorem to find the lengths of the (Given) metal supports. 2. (Opp. sides of a .)

PROOF Write a two-column proof. 3. (Refl. Prop.) 33. Theorem 6.13 4. are right angles. (Def. of rectangle) SOLUTION: 5. (All right .) You need to walk through the proof step by step. Look over what you are given and what you need to 6. (SAS) prove. Here, you are given WXYZ is a rectangle with 7. (CPCTC) diagonals . You need to prove . Use the properties that you have learned 34. Theorem 6.14 about rectangles to walk through the proof. SOLUTION:

You need to walk through the proof step by step. Given: WXYZ is a rectangle with diagonals Look over what you are given and what you need to

. prove. Here, you are given Prove: . You need to prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk through the proof.

Given: Proof: Prove: WXYZ is a rectangle. 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of

rectangle) Proof:

5. (All right .) 1. (Given) 6. (SAS) 2. (SSS)

7. (CPCTC) 3. (CPCTC) 4. (Def. of ) ANSWER: 5. WXYZ is a parallelogram. (If both pairs of opp. Given: WXYZ is a rectangle with diagonals sides are , then quad. is .) . 6. are supplementary. (Cons. Prove: are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1

Proof: rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) 1. WXYZ is a rectangle with diagonals . (Given) ANSWER: 2. (Opp. sides of a .) Given: 3. (Refl. Prop.) Prove: WXYZ is a rectangle. 4. are right angles. (Def. of rectangle) 5. (All right .) 6. (SAS)

7. (CPCTC) Proof: 1. (Given) 34. Theorem 6.14 2. (SSS) SOLUTION: 3. (CPCTC) You need to walk through the proof step by step. 4. (Def. of ) Look over what you are given and what you need to 5. WXYZ is a parallelogram. (If both pairs of opp. prove. Here, you are given sides are , then quad. is .) . You need to 6. are supplementary. (Cons. prove that WXYZ is a rectangle.. Use the properties are suppl.) that you have learned about rectangles to walk 7. (Def of suppl.) through the proof. 8. are right angles. (If

Given: 2 and suppl., each is a rt. .) Prove: WXYZ is a rectangle. 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a Proof: rectangle. 1. (Given) SOLUTION:

2. (SSS) You need to walk through the proof step by step. 3. (CPCTC) Look over what you are given and what you need to 4. (Def. of ) prove. Here, you are given a parallelogram with one 5. WXYZ is a parallelogram. (If both pairs of opp. right angle. You need to prove that it is a rectangle. sides are , then quad. is .) Let ABCD be a parallelogram with one right angle. 6. are supplementary. (Cons. Then use the properties of parallelograms to walk through the proof. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .)

10. WXYZ is a rectangle. (Def. of rectangle) ABCD is a parallelogram, and is a right angle. ANSWER: Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the

Given: definition of a rectangle, ABCD is a rectangle. Prove: WXYZ is a rectangle. ANSWER:

Proof:

1. (Given) ABCD is a parallelogram, and is a right angle. 2. (SSS) Since ABCD is a parallelogram and has one right 3. (CPCTC) angle, then it has four right angles. So by the 4. (Def. of ) definition of a rectangle, ABCD is a rectangle. 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 36. If a quadrilateral has four right angles, then it is a 6. are supplementary. (Cons. rectangle. are suppl.) SOLUTION: 7. (Def of suppl.) You need to walk through the proof step by step. 8. are right angles. (If Look over what you are given and what you need to prove. Here, you are given a quadrilateral with four 2 and suppl., each is a rt. .) right angles. You need to prove that it is a rectangle. 9. are right angles. (If a has 1 Let ABCD be a quadrilateral with four right angles. rt. , it has 4 rt. .) Then use the properties of parallelograms and 10. WXYZ is a rectangle. (Def. of rectangle) rectangles to walk through the proof.

PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to ABCD is a quadrilateral with four right angles. prove. Here, you are given a parallelogram with one ABCD is a parallelogram because both pairs of right angle. You need to prove that it is a rectangle. opposite angles are congruent. By definition of a Let ABCD be a parallelogram with one right angle. rectangle, ABCD is a rectangle. Then use the properties of parallelograms to walk ANSWER: through the proof.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of ABCD is a parallelogram, and is a right angle. opposite angles are congruent. By definition of a Since ABCD is a parallelogram and has one right rectangle, ABCD is a rectangle. angle, then it has four right angles. So by the

definition of a rectangle, ABCD is a rectangle. 37. CONSTRUCTION Construct a rectangle using the ANSWER: construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. SOLUTION: Sample answer: Step 1: Graph a line and place points P and Q on the ABCD is a parallelogram, and is a right angle. line. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the Step 2: Place the compass on point P. Using the definition of a rectangle, ABCD is a rectangle. same compass setting, draw an arc to the left and right of P that intersects . 36. If a quadrilateral has four right angles, then it is a rectangle. Step 3: Open the compass to a setting greater than SOLUTION: the distance from P to either point of intersection. You need to walk through the proof step by step. Place the compass on each point of intersection and Look over what you are given and what you need to draw arcs that intersect above . prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Step 4: Draw line a through point P and the Let ABCD be a quadrilateral with four right angles. intersection of the arcs drawn in step 3. Then use the properties of parallelograms and rectangles to walk through the proof. Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. ABCD is a quadrilateral with four right angles. Label the point of intersection R. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a Step 8: Using the same compass setting, put the rectangle, ABCD is a rectangle. compass at Q and draw an arc above it that intersects line b. Label the point of intersection S. ANSWER: Step 9: Draw .

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line Since and , the measure of through a point on the line. Justify each step of the angles P and Q is 90 degrees. Lines that are construction. perpendicular to the same line are parallel, so SOLUTION: . The same compass setting was used to Sample answer: locate points R and S, so . If one pair of Step 1: Graph a line and place points P and Q on the opposite sides of a quadrilateral is both parallel and

line. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Step 2: Place the compass on point P. Using the Thus, PRSQ is a rectangle. same compass setting, draw an arc to the left and right of P that intersects . ANSWER: Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Sample answer: Since and , the Place the compass on each point of intersection and measure of angles P and Q is 90 degrees. Lines that draw arcs that intersect above . are perpendicular to the same line are parallel, so . The same compass setting was used to Step 4: Draw line a through point P and the locate points R and S, so . If one pair of intersection of the arcs drawn in step 3. opposite sides of a quadrilateral is both parallel and

congruent, then the quadrilateral is a parallelogram. Step 5: Repeat steps 2 and 3 using point Q. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw . 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a Since and , the measure of parallelogram. If the diagonals are congruent, then angles P and Q is 90 degrees. Lines that are the end zone is a rectangle. perpendicular to the same line are parallel, so . The same compass setting was used to ANSWER: Sample answer: He should measure the diagonals of locate points R and S, so . If one pair of the end zone and both pairs of opposite sides. If the opposite sides of a quadrilateral is both parallel and diagonals and both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. congruent, then the end zone is a rectangle. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. ALGEBRA Quadrilateral WXYZ is a rectangle.

ANSWER:

Sample answer: Since and , the 39. If XW = 3, WZ = 4, and XZ = b, find YW. measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so SOLUTION: . The same compass setting was used to The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a locate points R and S, so . If one pair of rectangle are right angles. So, is a right opposite sides of a quadrilateral is both parallel and 2 2 congruent, then the quadrilateral is a parallelogram. triangle. By the Pythagorean Theorem, XZ = XW + 2 A parallelogram with right angles is a rectangle. WZ . Thus, PRSQ is a rectangle.

Therefore, YW = 5.

ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: The diagonals of a rectangle are congruent to each other. So, WY = XZ = 2c. All four angles of a 38. SPORTS The end zone of a football field is 160 feet rectangle are right angles. So, is a right wide and 30 feet long. Kyle is responsible for triangle. By the Pythagorean Theorem, XZ2 = XY2 + painting the field. He has finished the end zone. 2 Explain how Kyle can confirm that the end zone is ZY . the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both Therefore, WY = 2(5) = 10. pairs of opposite sides. If both pairs of opposite sides ANSWER: are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then 10 the end zone is a rectangle. 41. SIGNS The sign below is in the foyer of Nyoko’s ANSWER: school. Based on the dimensions given, can Nyoko Sample answer: He should measure the diagonals of be sure that the sign is a rectangle? Explain your the end zone and both pairs of opposite sides. If the reasoning. diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION:

SOLUTION: Both pairs of opposite sides are congruent, so the The diagonals of a rectangle are congruent to each sign is a parallelogram, but no measure is given that other. So, YW = XZ = b. All four angles of a can be used to prove that it is a rectangle. If the rectangle are right angles. So, is a right diagonals could be proven to be congruent, or if the triangle. By the Pythagorean Theorem, XZ2 = XW 2 + angles could be proven to be right angles, then we 2 could prove that the sign was a rectangle. WZ . ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no Therefore, YW = 5. measure is given that can be used to prove that it is a rectangle. ANSWER: 5 PROOF Write a coordinate proof of each statement. 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. 42. The diagonals of a rectangle are congruent. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent to each Begin by positioning quadrilateral ABCD on a other. So, WY = XZ = 2c. All four angles of a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b rectangle are right angles. So, is a right 2 2 units. Then the rest of the vertices are A(0, b), B(a, triangle. By the Pythagorean Theorem, XZ = XY + b), and C(a, 0). You need to walk through the proof 2 ZY . step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Therefore, WY = 2(5) = 10. Prove: ANSWER: 10

41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your reasoning.

Proof: Use the Distance Formula to find the lengths of the diagonals.

The diagonals, have the same length, so SOLUTION: they are congruent.

Both pairs of opposite sides are congruent, so the ANSWER: sign is a parallelogram, but no measure is given that Given: ABCD is a rectangle. can be used to prove that it is a rectangle. If the Prove: diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle.

ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a

rectangle. Proof:

PROOF Write a coordinate proof of each Use the Distance Formula to find statement. and . 42. The diagonals of a rectangle are congruent. have the same length, so they are SOLUTION: congruent. Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let 43. If the diagonals of a parallelogram are congruent, the length of the bases be a units and the height be b then it is a rectangle. units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof SOLUTION: step by step. Look over what you are given and what Begin by positioning parallelogram ABCD on a you need to prove. Here, you are given ABCD is a coordinate plane. Place vertex A at the origin. Let rectangle and you need to prove . Use the the length of the bases be a units and the height be c properties that you have learned about rectangles to units. Then the rest of the vertices are B(a, 0), C(b + walk through the proof. a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given Given: ABCD is a rectangle. and what you need to prove. Here, you are given Prove: ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Prove: ABCD is a rectangle.

Proof: Use the Distance Formula to find the lengths of the diagonals.

Proof:

The diagonals, have the same length, so they are congruent.

ANSWER: AC = BD. So, Given: ABCD is a rectangle. Prove:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined Proof:

Use the Distance Formula to find and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle. and . have the same length, so they are ANSWER: 6-4 Rectangles congruent. Given: ABCD and Prove: ABCD is a rectangle. 43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + Proof: a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given

ABCD and and you need to prove that But AC = BD. So, ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Prove: ABCD is a rectangle.

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.. Proof: 44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each AC = BD. So, with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals Because A and B are different points, of a parallelogram with four congruent sides. a ≠ 0. Then b = 0. The slope of is undefined SOLUTION: and the slope of . Thus, . ∠DAB is a. a right angle and ABCD is a rectangle. Sample answer:

ANSWER: Given: ABCD and Prove: ABCD is a rectangle.

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Proof:

b. Use a protractor to measure each angle listed in the table.

But AC = BD. So, c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: a. Sample answer: Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular. c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. 45. CHALLENGE In rectangle ABCD, SOLUTION: Find the a. Sample answer: values of x and y.

SOLUTION: All four angles of a rectangle are right angles. So,

b. Use a protractor to measure each angle listed in the table. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle c. Sample answer: Each of the angles listed in the with table are right angles. These angles were each So, formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: a. Sample answer: The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a b. rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning.

c. Sample answer: The diagonals of a parallelogram SOLUTION: with four congruent sides are perpendicular. When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a 45. CHALLENGE In rectangle ABCD, single vertex of a triangle. Find the values of x and y.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

SOLUTION: All four angles of a rectangle are right angles. So,

So, Tamika is correct.

The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle Tamika; Sample answer: When two congruent with triangles are arranged to form a quadrilateral, two of So, the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

47. REASONING In the diagram at the right, lines n, p, The sum of the angles of a triangle is 180. q,and r are parallel and lines l and m are parallel. So, How many rectangles are formed by the intersecting lines? By the Vertical Angle Theorem,

ANSWER: x = 6, y = −10 SOLUTION: 46. CCSS CRITIQUE Parker says that any two Let the lines n, p, q, and r intersect the line l at the congruent acute triangles can be arranged to make a points A, B, C, and D respectively. Similarly, let the rectangle. Tamika says that only two congruent right lines intersect the line m at the points W, X, Y, and Z. triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

In order for the quadrilateral to be a rectangle, one of Then the rectangles formed are ABXW, ACYW, the angles in the congruent triangles has to be a right BCYX, BDZX, CDZY, and ADZW. Therefore, 6 angle. rectangles are formed by the intersecting lines. ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. So, Tamika is correct. SOLUTION: ANSWER: Sample answer: Tamika; Sample answer: When two congruent A rectangle has 4 right angles and each pair of triangles are arranged to form a quadrilateral, two of opposite sides is parallel and congruent. Lines that the angles are formed by a single vertex of a triangle. run parallel to the x-axis will be parallel to each other In order for the quadrilateral to be a rectangle, one of and perpendicular to lines parallel to the y-axis. the angles in the congruent triangles has to be a right Selecting y = 0 and x = 0 will form 2 sides of the angle. rectangle. The lines y = 4 and x = 6 form the other 2 sides. 47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. x = 0, x = 6, y = 0, y = 4; How many rectangles are formed by the intersecting lines?

The length of is 6 − 0 or 6 units and the length of SOLUTION: Let the lines n, p, q, and r intersect the line l at the is 6 – 0 or 6 units. The slope of AB is 0 and the points A, B, C, and D respectively. Similarly, let the slope of DC is 0. Since a pair of sides of the lines intersect the line m at the points W, X, Y, and Z. quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a The length of is 6 − 0 or 6 units and the length of rectangle. Verify your answer using coordinate is 6 – 0 or 6 units. The slope of AB is 0 and the geometry. slope of DC is 0. Since a pair of sides of the SOLUTION: quadrilateral is both parallel and congruent, by Sample answer: Theorem 6.12, the quadrilateral is a parallelogram. A rectangle has 4 right angles and each pair of Since is horizontal and is vertical, the lines opposite sides is parallel and congruent. Lines that are perpendicular and the measure of the angle they run parallel to the x-axis will be parallel to each other form is 90. By Theorem 6.6, if a parallelogram has and perpendicular to lines parallel to the y-axis. one right angle, it has four right angles. Therefore, by Selecting y = 0 and x = 0 will form 2 sides of the definition, the parallelogram is a rectangle. rectangle. The lines y = 4 and x = 6 form the other 2 sides. 49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not x = 0, x = 6, y = 0, y = 4; rectangles. SOLUTION: Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms.

Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with The length of is 6 − 0 or 6 units and the length of non-right angles are not. is 6 – 0 or 6 units. The slope of AB is 0 and the ANSWER: slope of DC is 0. Since a pair of sides of the Sample answer: All rectangles are parallelograms quadrilateral is both parallel and congruent, by because, by definition, both pairs of opposite sides Theorem 6.12, the quadrilateral is a parallelogram. are parallel. Parallelograms with right angles are Since is horizontal and is vertical, the lines rectangles, so some parallelograms are rectangles, are perpendicular and the measure of the angle they but others with non-right angles are not. form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = definition, the parallelogram is a rectangle. 13, what values of x and y make parallelogram FGHJ a rectangle? ANSWER: Sample answer: x = 0, x = 6, y = 0, y = 4;

A x = 3, y = 4 B x = 4, y = 3 C x = 7, y = 8 D x = 8, y = 7 SOLUTION: The opposite sides of a parallelogram are congruent. So, FJ = GH. The length of is 6 − 0 or 6 units and the length of –3x + 5y = 11 FJ = GH is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the If the diagonals of a parallelogram are congruent, quadrilateral is both parallel and congruent, by then it is a rectangle. Since FGHJ is a parallelogram, Theorem 6.12, the quadrilateral is a parallelogram. the diagonals bisect each other. So, if it is a Since is horizontal and is vertical, the lines rectangle, then will be an isosceles triangle are perpendicular and the measure of the angle they with That is, 3x + y = 13. form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by Add the two equations to eliminate the x-term and definition, the parallelogram is a rectangle. solve for y. –3x + 5y = 11 49. WRITING IN MATH Explain why all rectangles 3x + y = 13 are parallelograms, but all parallelograms are not 6y = 24 Add to eliminate x-term. rectangles. y = 4 Divide each side by 6.

SOLUTION: Use the value of y to find the value of x. Sample answer: A parallelogram is a quadrilateral in 3x + 4 = 13 Substitute. which each pair of opposite sides are parallel. By 3x = 9 Subtract 4 from each side. definition of rectangle, all rectangles have both pairs x = 3 Divide each side by 3. of opposite sides parallel. So rectangles are always

parallelograms. So, x = 3, y = 4, the correct choice is A.

Parallelograms with right angles are rectangles, so ANSWER: some parallelograms are rectangles, but others with A non-right angles are not. 51. ALGEBRA A rectangular playground is surrounded ANSWER: by an 80-foot fence. One side of the playground is 10 Sample answer: All rectangles are parallelograms feet longer than the other. Which of the following because, by definition, both pairs of opposite sides equations could be used to find r, the shorter side of are parallel. Parallelograms with right angles are the playground? rectangles, so some parallelograms are rectangles, F 10r + r = 80 but others with non-right angles are not. G 4r + 10 = 80 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = H r(r + 10) = 80 13, what values of x and y make parallelogram J 2(r + 10) + 2r = 80 FGHJ a rectangle? SOLUTION: The formula to find the perimeter of a rectangle is given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. A x = 3, y = 4 Therefore, the correct choice is J. B x = 4, y = 3 ANSWER: C x = 7, y = 8 J D x = 8, y = 7 SOLUTION: 52. SHORT RESPONSE What is the measure of

The opposite sides of a parallelogram are congruent. ? So, FJ = GH. –3x + 5y = 11 FJ = GH

If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a rectangle, then will be an isosceles triangle SOLUTION: with That is, 3x + y = 13. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle Add the two equations to eliminate the x-term and with solve for y. So, –3x + 5y = 11 By the Exterior Angle Theorem, 3x + y = 13 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6. ANSWER:

112 Use the value of y to find the value of x.

3x + 4 = 13 Substitute. 53. SAT/ACT If p is odd, which of the following must

3x = 9 Subtract 4 from each side. also be odd?

x = 3 Divide each side by 3. A 2p

B 2p + 2 So, x = 3, y = 4, the correct choice is A.

C ANSWER: A D E p + 2 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 SOLUTION: feet longer than the other. Which of the following Analyze each answer choice to determine which equations could be used to find r, the shorter side of expression is odd when p is odd. the playground? Choice A 2p F 10r + r = 80 This is a multiple of 2 so it is even. G 4r + 10 = 80 Choice B 2p + 2 H r(r + 10) = 80 This can be rewritten as 2(p + 1) which is also a J 2(r + 10) + 2r = 80 multiple of 2 so it is even.

SOLUTION: Choice C The formula to find the perimeter of a rectangle is If p is odd , is not a whole number. given by P = 2(l + w). The perimeter is given to be 8 Choice D 2p – 2 ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. This can be rewritten as 2(p - 1) which is also a Therefore, the correct choice is J. multiple of 2 so it is even. Choice E p + 2 ANSWER: Adding 2 to an odd number results in an odd J number. This is odd. The correct choice is E. 52. SHORT RESPONSE What is the measure of ? ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

SOLUTION: The diagonals of a rectangle are congruent and 54. bisect each other. So, is an isosceles triangle with SOLUTION: So, The opposite sides of a parallelogram are congruent. By the Exterior Angle Theorem,

ANSWER: In a parallelogram, consecutive interior angles are 112 supplementary.

53. SAT/ACT If p is odd, which of the following must also be odd? A 2p B 2p + 2

C ANSWER: D x = 2, y = 41 E p + 2 SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 55. This can be rewritten as 2(p + 1) which is also a SOLUTION: multiple of 2 so it is even. The opposite sides of a parallelogram are congruent.

Choice C If p is odd , is not a whole number. Choice D 2p – 2 The opposite sides of a parallelogram are parallel to This can be rewritten as 2(p - 1) which is also a each other. So, the angles with the measures (3y + multiple of 2 so it is even. 3)° and (4y – 19)° are alternate interior angles and Choice E p + 2 hence they are equal. Adding 2 to an odd number results in an odd number. This is odd. The correct choice is E.

ANSWER: ANSWER: E x = 8, y = 22 ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

56. 54. SOLUTION: The diagonals of a parallelogram bisect each other. SOLUTION: So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. The opposite sides of a parallelogram are congruent. Solve the first equation to find the value of y. 2y = 14 y = 7 Use the value of y in the second equation and solve In a parallelogram, consecutive interior angles are for x. supplementary. 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: x = 2, y = 7

57. COORDINATE GEOMETRY Find the ANSWER: coordinates of the intersection of the diagonals of x = 2, y = 41 parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or . 55.

SOLUTION: The opposite sides of a parallelogram are congruent. Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5). ANSWER: (2.5, 0.5) The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + Refer to the figure. 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

ANSWER: x = 8, y = 22 58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

56. SOLUTION: ANSWER: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6.

Solve the first equation to find the value of y. 59. If , name two congruent segments. 2y = 14 y = 7 SOLUTION: Use the value of y in the second equation and solve In a triangle with two congruent angles, the sides for x. opposite to the congruent angles are also congruent. 7 + 3 = 2x + 6 So, . 4 = 2x 2 = x ANSWER: ANSWER: x = 2, y = 7 60. If , name two congruent segments. 57. COORDINATE GEOMETRY Find the SOLUTION: coordinates of the intersection of the diagonals of In a triangle with two congruent angles, the sides parallelogram ABCD with vertices A(1, 3), B(6, 2), C opposite to the congruent angles are also congruent. (4, –2), and D(–1, –1). So, . SOLUTION: The diagonals of a parallelogram bisect each other, ANSWER: so the intersection is the midpoint of either diagonal. Find the midpoint of or .

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps ANSWER: to prevent the door from warping over time. If feet, PS = 7 feet, and

, find each measure. 3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Therefore,

ANSWER: 1. QR 33.5 SOLUTION: The opposite sides of a rectangle are parallel and 4. congruent. Therefore, QR = PS = 7 ft. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 7 ft bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem, 2. SQ SOLUTION: Therefore, The diagonals of a rectangle bisect each other. So,

ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

ANSWER: 5. If FD = 3x – 7 and EG = x + 5, find EG.

SOLUTION: The diagonals of a rectangle are congruent to each 3. other. So, FD = EG.

ANSWER: 6. If , 33.5 find . SOLUTION: 4. All four angles of a rectangle are right angles. So, SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore, ANSWER: ANSWER: 51 56.5 7. PROOF If ABDE is a rectangle and , ALGEBRA Quadrilateral DEFG is a rectangle. prove that .

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each SOLUTION:

other. So, FD = EG. You need to walk through the proof step by step. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and . You need to prove . Use the Use the value of x to find EG. properties that you have learned about rectangles to EG = 6 + 5 = 11 walk through the proof.

ANSWER: Given: ABDE is a rectangle; 11 Prove: 6. If , find . SOLUTION: All four angles of a rectangle are right angles. So,

Given: ABDE is a rectangle; 3. (Opp. sides of a .) Prove: 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your Statements(Reasons) answer using the indicated formula. 1. ABDE is a rectangle; . 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula (Given) SOLUTION: 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle;

Prove: Use the slope formula to find the slope of the sides of the quadrilateral.

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) ANSWER: 5. (All rt .) No; slope of , slope of , slope of 6. (SAS)

7. (CPCTC) , and slope of . Slope of

answer using the indicated formula. WXYZ is not a rectangle. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula Use the slope formula to find the slope of the sides of the quadrilateral. SOLUTION:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. First, Use the Distance formula to find the lengths of ANSWER: the sides of the quadrilateral.

No; slope of , slope of , slope of

, and slope of . Slope of

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

So the diagonals are congruent. Thus, ABCD is a ANSWER:

rectangle. 2 ft ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a 11. DB parallelogram. , so the diagonals are SOLUTION: congruent. Thus, ABCD is a rectangle. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

ANSWER: 6.3 ft

By the Vertical Angle Theorem,

SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . ANSWER: BC = AD = 2 25 DC = AB = 6 CCSS REGULARITY Quadrilateral WXYZ is a rectangle.

ANSWER: 6.3 ft

Use the value of x to find WX. WX = 1 + 4 = 5.

By the Vertical Angle Theorem, ANSWER: 5 ANSWER: 50 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION:

13. The diagonals of a rectangle bisect each other. So, SOLUTION: PY = WP. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. The diagonals of a rectangle are congruent and Then, bisect each other. So, is an isosceles triangle. All four angles of a rectangle are right angles.

ANSWER: ANSWER: 43 25 16. If , find CCSS REGULARITY Quadrilateral WXYZ is a . rectangle. SOLUTION: All four angles of a rectangle are right angles. So,

14. If ZY = 2x + 3 and WX = x + 4, find WX.

15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: Then ZP = 4(7) – 9 = 19. The diagonals of a rectangle bisect each other. So, Therefore, ZX = 2(ZP) = 38. PY = WP. ANSWER: 38 The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 18. If , find . SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 43

16. If , find . SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 48

19. If , find . SOLUTION:

All four angles of a rectangle are right angles. So, is a right triangle. Then,

ANSWER: Then ZP = 4(7) – 9 = 19. 46 Therefore, ZX = 2(ZP) = 38. PROOF Write a two-column proof. ANSWER: 20. Given: ABCD is a rectangle. 38 Prove:

18. If , find . SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION:

You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof. ANSWER: 48 Given: ABCD is a rectangle. Prove: 19. If , find Proof: . Statements (Reasons) 1. ABCD is a rectangle. (Given) SOLUTION: 2. ABCD is a parallelogram. (Def. of rectangle) All four angles of a rectangle are right angles. So, 3. (Opp. sides of a .) is a right triangle. Then, 4. (Refl. Prop.)

5. (Diagonals of a rectangle are .) 6. (SSS)

ANSWER: Therefore, Since is a Proof: Statements (Reasons) transversal of parallel sides and , alternate 1. ABCD is a rectangle. (Given) interior angles WZX and ZXY are congruent. So, 2. ABCD is a parallelogram. (Def. of rectangle)

= = 46. 3. (Opp. sides of a .) ANSWER: 4. (Refl. Prop.) 46 5. (Diagonals of a rectangle are .) 6. (SSS) PROOF Write a two-column proof. 20. Given: ABCD is a rectangle. Prove: 21. Given: QTVW is a rectangle.

the properties that you have learned about rectangles Given: ABCD is a rectangle. to walk through the proof.

Prove:

Proof: Given: QTVW is a rectangle.

Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS) Prove: Proof: ANSWER: Statements (Reasons) Proof: 1. QTVW is a rectangle; .(Given) Statements (Reasons) 2. QTVW is a parallelogram. (Def. of rectangle) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 3. (Opp. sides of a .) 5. (All rt .) 4. (Refl. Prop.)

6. QR = ST (Def. of segs.) 5. (Diagonals of a rectangle are .) 6. (SSS) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 21. Given: QTVW is a rectangle. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Prove: Statements (Reasons) SOLUTION: 1. QTVW is a rectangle; .(Given) You need to walk through the proof step by step. 2. QTVW is a parallelogram. (Def. of rectangle) Look over what you are given and what you need to 3. (Opp sides of a .) prove. Here, you are given QTVW is a rectangle and 4. are right angles.(Def. of rectangle) . You need to prove . Use 5. (All rt .) the properties that you have learned about rectangles

to walk through the proof. 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) Given: QTVW is a rectangle. 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

Prove: COORDINATE GEOMETRY Graph each Proof: quadrilateral with the given vertices. Determine Statements (Reasons) whether the figure is a rectangle. Justify your 1. QTVW is a rectangle; .(Given) answer using the indicated formula. 2. QTVW is a parallelogram. (Def. of rectangle) 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula 3. (Opp sides of a .) SOLUTION: 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.)

13. (SAS) Use the slope formula to find the slope of the sides of ANSWER: the quadrilateral. Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) The slopes of each pair of opposite sides are equal. 8. RS = RS (Def. of segs.) So, the two pairs of opposite sides are parallel. 9. QR + RS = RS + ST (Add. prop.) Therefore, the quadrilateral WXYZ is a parallelogram. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) The products of the slopes of the adjacent sides are 11. QS = RT (Subst.) –1. So, any two adjacent sides are perpendicular to 12. (Def. of segs.) each other. That is, all the four angles are right 13. (SAS) angles. Therefore, WXYZ is a rectangle. ANSWER: COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine Yes; slope of slope of , slope whether the figure is a rectangle. Justify your answer using the indicated formula. of slope of . So WXYZ is a 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula parallelogram. The product of the slopes of SOLUTION: consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

ANSWER: First, Use the Distance formula to find the lengths of the sides of the quadrilateral. Yes; slope of slope of , slope

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance rectangle. Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula

First, Use the Distance formula to find the lengths of SOLUTION: the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

First, use the Distance Formula to find the lengths of the sides of the quadrilateral. So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a so QRST is a parallelogram; so parallelogram. the diagonals are not congruent. Thus, JKLM is not a A parallelogram is a rectangle if the diagonals are

rectangle. congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the

lengths of the diagonals. First, use the slope formula to find the lengths of the sides of the quadrilateral. So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

ANSWER:

No; slope of slope of and slope of 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula slope of . So, GHJK is a SOLUTION: parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

First, use the slope formula to find the lengths of the sides of the quadrilateral.

Quadrilateral ABCD is a rectangle. Find each measure if .

26. The slopes of each pair of opposite sides are equal. SOLUTION: So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a All four angles of a rectangle are right angles. So, parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle. ANSWER: ANSWER: 50 No; slope of slope of and slope of 27. slope of . So, GHJK is a parallelogram. The product of the slopes of SOLUTION: consecutive sides ≠ –1, so the consecutive sides are The measures of angles 2 and 3 are equal as they not perpendicular. Thus, GHJK is not a rectangle. are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: 40

28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. Quadrilateral ABCD is a rectangle. Find each ANSWER: measure if . 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

26. SOLUTION: All four angles of a rectangle are right angles. So, The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 ANSWER: and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. 50 The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. 27. ANSWER: SOLUTION: 80 The measures of angles 2 and 3 are equal as they 30. are alternate interior angles. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 All four angles of a rectangle are right angles. So, and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: The measures of angles 1 and 4 are equal as they 40 are alternate interior angles. 28. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 The measures of angles 2 and 3 are equal as they and 6 is an isosceles triangle. Therefore, m∠6 = are alternate interior angles. m∠4 = 50. Therefore, m∠3 = m∠2 = 40. ANSWER: ANSWER: 50 40 31. 29. SOLUTION: SOLUTION: The measures of angles 2 and 3 are equal as they All four angles of a rectangle are right angles. So, are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The measures of angles 1 and 4 are equal as they The sum of the three angles of a triangle is 180. are alternate interior angles. Therefore, m∠8 = 180 – (40 + 40) = 100.

Therefore, m∠4 = m∠1 = 50. ANSWER: Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 100

and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. 32. CCSS MODELING Jody is building a new

The sum of the three angles of a triangle is 180. bookshelf using wood and metal supports like the one Therefore, m∠5 = 180 – (50 + 50) = 80. shown. To what length should she cut the metal supports in order for the bookshelf to be square, ANSWER: which means that the angles formed by the shelves 80 and the vertical supports are all right angles? Explain your reasoning. 30. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. SOLUTION: Since the diagonals of a rectangle are congruent and In order for the bookshelf to be square, the lengths of bisect each other, the triangle with the angles 4, 5 all of the metal supports should be equal. Since we and 6 is an isosceles triangle. Therefore, m∠6 = know the length of the bookshelves and the distance m∠4 = 50. between them, use the Pythagorean Theorem to find the lengths of the metal supports. ANSWER: Let x be the length of each support. Express 3 feet 50 as 36 inches.

31. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and Therefore, the metal supports should each be 39 bisect each other, the triangle with the angles 3, 7 inches long or 3 feet 3 inches long. and 8 is an isosceles triangle. So, m 3 = m 7 = 40. ∠ ∠ ANSWER: The sum of the three angles of a triangle is 180. 3 ft 3 in.; Sample answer: In order for the bookshelf Therefore, m∠8 = 180 – (40 + 40) = 100. to be square, the lengths of all of the metal supports ANSWER: should be equal. Since I knew the length of the 100 bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the 32. CCSS MODELING Jody is building a new metal supports. bookshelf using wood and metal supports like the one PROOF Write a two-column proof. shown. To what length should she cut the metal 33. Theorem 6.13 supports in order for the bookshelf to be square, which means that the angles formed by the shelves SOLUTION: and the vertical supports are all right angles? Explain You need to walk through the proof step by step. your reasoning. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: WXYZ is a rectangle with diagonals

SOLUTION: Prove: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find the lengths of the metal supports. Let x be the length of each support. Express 3 feet Proof: as 36 inches. 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of Therefore, the metal supports should each be 39 rectangle) inches long or 3 feet 3 inches long. 5. (All right .) 6. (SAS) ANSWER:

3 ft 3 in.; Sample answer: In order for the bookshelf 7. (CPCTC) to be square, the lengths of all of the metal supports ANSWER: should be equal. Since I knew the length of the bookshelves and the distance between them, I used Given: WXYZ is a rectangle with diagonals the Pythagorean Theorem to find the lengths of the . metal supports. Prove: PROOF Write a two-column proof. 33. Theorem 6.13 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to Proof: prove. Here, you are given WXYZ is a rectangle with 1. WXYZ is a rectangle with diagonals . diagonals . You need to prove (Given) . Use the properties that you have learned 2. (Opp. sides of a .) about rectangles to walk through the proof. 3. (Refl. Prop.)

4. are right angles. (Def. of Given: WXYZ is a rectangle with diagonals

rectangle) . 5. (All right .)

Prove: 6. (SAS) 7. (CPCTC)

34. Theorem 6.14

Proof: SOLUTION: You need to walk through the proof step by step. 1. WXYZ is a rectangle with diagonals . Look over what you are given and what you need to (Given) prove. Here, you are given

2. (Opp. sides of a .) . You need to 3. (Refl. Prop.) prove that WXYZ is a rectangle.. Use the properties 4. are right angles. (Def. of that you have learned about rectangles to walk rectangle) through the proof.

5. (All right .)

6. (SAS) Given: Prove: WXYZ is a rectangle. 7. (CPCTC)

ANSWER: Given: WXYZ is a rectangle with diagonals .

Prove: Proof: 1. (Given) 2. (SSS) 3. (CPCTC) 4. (Def. of ) Proof: 5. WXYZ is a parallelogram. (If both pairs of opp. 1. WXYZ is a rectangle with diagonals . sides are , then quad. is .) (Given) 6. are supplementary. (Cons. 2. (Opp. sides of a .) are suppl.) 3. (Refl. Prop.) 7. (Def of suppl.) 4. are right angles. (Def. of 8. are right angles. (If rectangle) 2 and suppl., each is a rt. .) 5. (All right .) 9. are right angles. (If a has 1 6. (SAS) rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) 7. (CPCTC) ANSWER: 34. Theorem 6.14 Given: SOLUTION: Prove: WXYZ is a rectangle. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given . You need to prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk Proof: through the proof. 1. (Given) 2. (SSS) Given: 3. (CPCTC) Prove: WXYZ is a rectangle. 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) Proof: 8. are right angles. (If 1. (Given) 2 and suppl., each is a rt. .) 2. (SSS) 9. are right angles. (If a has 1 3. (CPCTC) rt. , it has 4 rt. .) 4. (Def. of ) 10. WXYZ is a rectangle. (Def. of rectangle) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) PROOF Write a paragraph proof of each 6. are supplementary. (Cons. statement. are suppl.) 35. If a parallelogram has one right angle, then it is a rectangle. 7. (Def of suppl.) 8. are right angles. (If SOLUTION: 2 and suppl., each is a rt. .) You need to walk through the proof step by step. Look over what you are given and what you need to 9. are right angles. (If a has 1 prove. Here, you are given a parallelogram with one rt. , it has 4 rt. .) right angle. You need to prove that it is a rectangle. 10. WXYZ is a rectangle. (Def. of rectangle) Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk ANSWER: through the proof. Given: Prove: WXYZ is a rectangle.

10. WXYZ is a rectangle. (Def. of rectangle) 36. If a quadrilateral has four right angles, then it is a

PROOF Write a paragraph proof of each rectangle. statement. SOLUTION: 35. If a parallelogram has one right angle, then it is a You need to walk through the proof step by step. rectangle. Look over what you are given and what you need to SOLUTION: prove. Here, you are given a quadrilateral with four You need to walk through the proof step by step. right angles. You need to prove that it is a rectangle. Look over what you are given and what you need to Let ABCD be a quadrilateral with four right angles. prove. Here, you are given a parallelogram with one Then use the properties of parallelograms and

right angle. You need to prove that it is a rectangle. rectangles to walk through the proof.

Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

ANSWER:

definition of a rectangle, ABCD is a rectangle. construction for a line perpendicular to another line through a point on the line. Justify each step of the 36. If a quadrilateral has four right angles, then it is a construction. rectangle. SOLUTION: SOLUTION: Sample answer: You need to walk through the proof step by step. Step 1: Graph a line and place points P and Q on the Look over what you are given and what you need to line. prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Step 2: Place the compass on point P. Using the Let ABCD be a quadrilateral with four right angles. same compass setting, draw an arc to the left and Then use the properties of parallelograms and rectangles to walk through the proof. right of P that intersects .

Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Place the compass on each point of intersection and draw arcs that intersect above .

ANSWER: Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and Since and , the measure of right of P that intersects . angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 3: Open the compass to a setting greater than . The same compass setting was used to the distance from P to either point of intersection. Place the compass on each point of intersection and locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and draw arcs that intersect above . congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Step 4: Draw line a through point P and the Thus, PRSQ is a rectangle. intersection of the arcs drawn in step 3.

Step 5: Repeat steps 2 and 3 using point Q. ANSWER:

Step 6: Draw line b through point Q and the Sample answer: Since and , the intersection of the arcs drawn in step 5. measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. . The same compass setting was used to Label the point of intersection R. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and Step 8: Using the same compass setting, put the congruent, then the quadrilateral is a parallelogram. compass at Q and draw an arc above it that A parallelogram with right angles is a rectangle. intersects line b. Label the point of intersection S. Thus, PRSQ is a rectangle.

Step 9: Draw .

38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for Since and , the measure of painting the field. He has finished the end zone. angles P and Q is 90 degrees. Lines that are Explain how Kyle can confirm that the end zone is perpendicular to the same line are parallel, so the regulation size and be sure that it is also a rectangle using only a tape measure. . The same compass setting was used to SOLUTION: locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and Sample answer: If the diagonals of a parallelogram congruent, then the quadrilateral is a parallelogram. are congruent, then it is a rectangle. He should A parallelogram with right angles is a rectangle. measure the diagonals of the end zone and both Thus, PRSQ is a rectangle. pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then ANSWER: the end zone is a rectangle. ANSWER: Sample answer: Since and , the Sample answer: He should measure the diagonals of measure of angles P and Q is 90 degrees. Lines that the end zone and both pairs of opposite sides. If the are perpendicular to the same line are parallel, so diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. . The same compass setting was used to locate points R and S, so . If one pair of ALGEBRA Quadrilateral WXYZ is a rectangle. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

38. SPORTS The end zone of a football field is 160 feet Therefore, YW = 5. wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. ANSWER: Explain how Kyle can confirm that the end zone is 5 the regulation size and be sure that it is also a rectangle using only a tape measure. 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: SOLUTION: Sample answer: If the diagonals of a parallelogram The diagonals of a rectangle are congruent to each are congruent, then it is a rectangle. He should other. So, WY = XZ = 2c. All four angles of a measure the diagonals of the end zone and both rectangle are right angles. So, is a right pairs of opposite sides. If both pairs of opposite sides 2 2 are congruent, then the quadrilateral is a triangle. By the Pythagorean Theorem, XZ = XY + 2 parallelogram. If the diagonals are congruent, then ZY . the end zone is a rectangle. ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. Therefore, WY = 2(5) = 10.

ALGEBRA Quadrilateral WXYZ is a rectangle. ANSWER: 10

SOLUTION:

Therefore, YW = 5. Both pairs of opposite sides are congruent, so the ANSWER: sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the 5 diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. could prove that the sign was a rectangle. SOLUTION: ANSWER: The diagonals of a rectangle are congruent to each No; sample answer: Both pairs of opposite sides are other. So, WY = XZ = 2c. All four angles of a congruent, so the sign is a parallelogram, but no rectangle are right angles. So, is a right measure is given that can be used to prove that it is a triangle. By the Pythagorean Theorem, XZ2 = XY2 + rectangle. 2 ZY . PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let Therefore, WY = 2(5) = 10. the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, ANSWER: b), and C(a, 0). You need to walk through the proof 10 step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a 41. SIGNS The sign below is in the foyer of Nyoko’s rectangle and you need to prove . Use the school. Based on the dimensions given, can Nyoko properties that you have learned about rectangles to be sure that the sign is a rectangle? Explain your walk through the proof. reasoning.

Given: ABCD is a rectangle. Prove:

SOLUTION:

PROOF Write a coordinate proof of each Prove: statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof Proof: step by step. Look over what you are given and what Use the Distance Formula to find you need to prove. Here, you are given ABCD is a and . rectangle and you need to prove . Use the have the same length, so they are properties that you have learned about rectangles to congruent. walk through the proof.

Given: ABCD is a rectangle. 43. If the diagonals of a parallelogram are congruent, Prove: then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given Proof: Use the Distance Formula to find the lengths of the ABCD and and you need to prove that diagonals. ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and The diagonals, have the same length, so Prove: ABCD is a rectangle. they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove:

Proof:

AC = BD. So, Proof: Use the Distance Formula to find and . have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined SOLUTION: Begin by positioning parallelogram ABCD on a and the slope of . Thus, . ∠DAB is coordinate plane. Place vertex A at the origin. Let a right angle and ABCD is a rectangle. the length of the bases be a units and the height be c ANSWER: units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the Given: ABCD and proof step by step. Look over what you are given Prove: ABCD is a rectangle. and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Proof: Prove: ABCD is a rectangle.

But AC = BD. So,

Proof:

AC = BD. So, Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined

and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. Because A and B are different points, a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one a ≠ 0. Then b = 0. The slope of is undefined parallelogram ABCD, one MNOP, and one WXYZ. and the slope of . Thus, . ∠DAB is Draw the two diagonals of each parallelogram and a right angle and ABCD is a rectangle. label the intersections R. b. TABULAR Use a protractor to measure the ANSWER: appropriate angles and complete the table below. Given: ABCD and Prove: ABCD is a rectangle. c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

Proof:

But AC = BD. So,

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer: b.

6-4 Rectangles c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

b. Use a protractor to measure each angle listed in the table.

The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

values of x and y. congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

eSolutionsSOLUTION: Manual - Powered by Cognero Page 17 All four angles of a rectangle are right angles. So,

So, Tamika is correct. The sum of the angles of a triangle is 180. So, ANSWER: By the Vertical Angle Theorem, Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

ANSWER: 47. REASONING In the diagram at the right, lines n, p, x = 6, y = −10 q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting 46. CCSS CRITIQUE Parker says that any two lines? congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

47. REASONING In the diagram at the right, lines n, p, geometry. q,and r are parallel and lines l and m are parallel. SOLUTION: How many rectangles are formed by the intersecting Sample answer: lines? A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines Then the rectangles formed are ABXW, ACYW, are perpendicular and the measure of the angle they BCYX, BDZX, CDZY, and ADZW. Therefore, 6 form is 90. By Theorem 6.6, if a parallelogram has rectangles are formed by the intersecting lines. one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. ANSWER: 6 ANSWER: Sample answer: x = 0, x = 6, y = 0, OPEN ENDED 48. Write the equations of four lines y = 4; having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the The length of is 6 − 0 or 6 units and the length of rectangle. The lines y = 4 and x = 6 form the other 2 sides. is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the x = 0, x = 6, y = 0, y = 4; quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram FGHJ a rectangle?

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the A x = 3, y = 4 quadrilateral is both parallel and congruent, by B x = 4, y = 3 Theorem 6.12, the quadrilateral is a parallelogram. C x = 7, y = 8 Since is horizontal and is vertical, the lines D x = 8, y = 7 are perpendicular and the measure of the angle they SOLUTION: form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by The opposite sides of a parallelogram are congruent. definition, the parallelogram is a rectangle. So, FJ = GH. –3x + 5y = 11 FJ = GH 49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not If the diagonals of a parallelogram are congruent, rectangles. then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a SOLUTION: rectangle, then will be an isosceles triangle Sample answer: A parallelogram is a quadrilateral in with That is, 3x + y = 13. which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs Add the two equations to eliminate the x-term and of opposite sides parallel. So rectangles are always solve for y. parallelograms. –3x + 5y = 11 3x + y = 13 Parallelograms with right angles are rectangles, so 6y = 24 Add to eliminate x-term. some parallelograms are rectangles, but others with y = 4 Divide each side by 6.

non-right angles are not. ANSWER: Use the value of y to find the value of x. 3x + 4 = 13 Substitute. Sample answer: All rectangles are parallelograms 3x = 9 Subtract 4 from each side. because, by definition, both pairs of opposite sides

are parallel. Parallelograms with right angles are x = 3 Divide each side by 3.

rectangles, so some parallelograms are rectangles,

but others with non-right angles are not. So, x = 3, y = 4, the correct choice is A. ANSWER: 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram A FGHJ a rectangle? 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following equations could be used to find r, the shorter side of

the playground? A x = 3, y = 4 F 10r + r = 80 B x = 4, y = 3 G 4r + 10 = 80 C x = 7, y = 8 H r(r + 10) = 80 D x = 8, y = 7 J 2(r + 10) + 2r = 80 SOLUTION: SOLUTION: The opposite sides of a parallelogram are congruent. The formula to find the perimeter of a rectangle is So, FJ = GH. given by P = 2(l + w). The perimeter is given to be 8 –3x + 5y = 11 FJ = GH ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. If the diagonals of a parallelogram are congruent, Therefore, the correct choice is J. then it is a rectangle. Since FGHJ is a parallelogram, ANSWER: the diagonals bisect each other. So, if it is a J rectangle, then will be an isosceles triangle with That is, 3x + y = 13. 52. SHORT RESPONSE What is the measure of ? Add the two equations to eliminate the x-term and solve for y. –3x + 5y = 11 3x + y = 13 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6.

So,

So, x = 3, y = 4, the correct choice is A. By the Exterior Angle Theorem, ANSWER: A ANSWER: 51. ALGEBRA A rectangular playground is surrounded 112 by an 80-foot fence. One side of the playground is 10 53. SAT/ACT If p is odd, which of the following must feet longer than the other. Which of the following also be odd? equations could be used to find r, the shorter side of the playground? A 2p F 10r + r = 80 B 2p + 2

G 4r + 10 = 80 C H r(r + 10) = 80 J 2(r + 10) + 2r = 80 D E p + 2 SOLUTION: The formula to find the perimeter of a rectangle is SOLUTION: given by P = 2(l + w). The perimeter is given to be 8 Analyze each answer choice to determine which ft. The width is r and the length is r + 10. So, 2r + 2 expression is odd when p is odd. (r + 10) = 80. Choice A 2p Therefore, the correct choice is J. This is a multiple of 2 so it is even. Choice B 2p + 2 ANSWER: This can be rewritten as 2(p + 1) which is also a J multiple of 2 so it is even.

52. SHORT RESPONSE What is the measure of Choice C ? If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. SOLUTION: The correct choice is E. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle ANSWER: with E So, By the Exterior Angle Theorem, ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

ANSWER: 112

53. SAT/ACT If p is odd, which of the following must 54. also be odd? A 2p SOLUTION: B 2p + 2 The opposite sides of a parallelogram are congruent.

C D E p + 2 In a parallelogram, consecutive interior angles are supplementary. SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a ANSWER: multiple of 2 so it is even. x = 2, y = 41

ANSWER: E The opposite sides of a parallelogram are parallel to ALGEBRA Find x and y so that the each other. So, the angles with the measures (3y + quadrilateral is a parallelogram. 3)° and (4y – 19)° are alternate interior angles and hence they are equal.

54. ANSWER: SOLUTION:

The opposite sides of a parallelogram are congruent. x = 8, y = 22

In a parallelogram, consecutive interior angles are supplementary. 56.

SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 ANSWER: y = 7 Use the value of y in the second equation and solve

x = 2, y = 41 for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: x = 2, y = 7 55. 57. COORDINATE GEOMETRY Find the SOLUTION: coordinates of the intersection of the diagonals of The opposite sides of a parallelogram are congruent. parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: The diagonals of a parallelogram bisect each other, The opposite sides of a parallelogram are parallel to so the intersection is the midpoint of either diagonal. each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and Find the midpoint of or . hence they are equal.

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: ANSWER: x = 8, y = 22 (2.5, 0.5) Refer to the figure.

ANSWER: x = 2, y = 7 59. If , name two congruent segments. 57. COORDINATE GEOMETRY Find the SOLUTION: coordinates of the intersection of the diagonals of In a triangle with two congruent angles, the sides parallelogram ABCD with vertices A(1, 3), B(6, 2), C opposite to the congruent angles are also congruent. (4, –2), and D(–1, –1). So, . SOLUTION: ANSWER: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or . 60. If , name two congruent segments. SOLUTION:

Therefore, the coordinates of the intersection of the In a triangle with two congruent angles, the sides diagonals are (2.5, 0.5). opposite to the congruent angles are also congruent. So, . ANSWER: (2.5, 0.5) ANSWER:

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and ANSWER: , find each measure.

3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

1. QR Therefore, SOLUTION: The opposite sides of a rectangle are parallel and ANSWER: congruent. Therefore, QR = PS = 7 ft. 33.5

ANSWER: 4. 7 ft SOLUTION: The diagonals of a rectangle are congruent and 2. SQ bisect each other. So, is an isosceles triangle. SOLUTION: Then, The diagonals of a rectangle bisect each other. So, By the Vertical Angle Theorem,

Therefore,

ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

ANSWER:

3. 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: SOLUTION: The diagonals of a rectangle are congruent to each The diagonals of a rectangle are congruent and other. So, FD = EG. bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Use the value of x to find EG.

Therefore, EG = 6 + 5 = 11 ANSWER: ANSWER: 11 33.5

4. 6. If , find . SOLUTION: The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, is an isosceles triangle. All four angles of a rectangle are right angles. So, Then, By the Vertical Angle Theorem,

Therefore,

ANSWER: 56.5 ANSWER: ALGEBRA Quadrilateral DEFG is a rectangle. 51

7. PROOF If ABDE is a rectangle and , prove that .

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

SOLUTION: You need to walk through the proof step by step. Use the value of x to find EG. Look over what you are given and what you need to EG = 6 + 5 = 11 prove. You are given ABDE is a rectangle and ANSWER: . You need to prove . Use the properties that you have learned about rectangles to 11 walk through the proof.

6. If , Given: ABDE is a rectangle; find . Prove: SOLUTION: All four angles of a rectangle are right angles. So,

Statements(Reasons) 1. ABDE is a rectangle; . (Given) ANSWER: 2. ABDE is a parallelogram. (Def. of 51 rectangle) 3. (Opp. sides of a .) 7. PROOF If ABDE is a rectangle and , 4. are right angles.(Def. of rectangle)

prove that . 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and . You need to prove . Use the properties that you have learned about rectangles to walk through the proof. Statements(Reasons) 1. ABDE is a rectangle; . Given: ABDE is a rectangle; (Given) 2. ABDE is a parallelogram. (Def. of Prove: rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

Statements(Reasons) COORDINATE GEOMETRY Graph each 1. ABDE is a rectangle; . quadrilateral with the given vertices. Determine (Given) whether the figure is a rectangle. Justify your 2. ABDE is a parallelogram. (Def. of answer using the indicated formula. rectangle) 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula

3. (Opp. sides of a .) SOLUTION: 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

Use the slope formula to find the slope of the sides of the quadrilateral.

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) There are no parallel sides, so WXYZ is not a 5. (All rt .) parallelogram. Therefore, WXYZ is not a rectangle. 6. (SAS) ANSWER: 7. (CPCTC) No; slope of , slope of , slope of COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your , and slope of . Slope of answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula slope of , and slope of slope of , so WXYZ is not a parallelogram. Therefore, SOLUTION: WXYZ is not a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral. 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of First, Use the Distance formula to find the lengths of the sides of the quadrilateral. , and slope of . Slope of

slope of , and slope of slope of , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a 9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance parallelogram. , so the diagonals are Formula congruent. Thus, ABCD is a rectangle. SOLUTION:

FENCING X-braces are also used to provide First, Use the Distance formula to find the lengths of support in rectangular fencing. If AB = 6 feet, the sides of the quadrilateral. AD = 2 feet, and , find each measure.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

10. BC

So the diagonals are congruent. Thus, ABCD is a SOLUTION: rectangle. The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a ANSWER: 2 ft parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. 11. DB SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

FENCING X-braces are also used to provide ANSWER: support in rectangular fencing. If AB = 6 feet, 6.3 ft AD = 2 feet, and , find each

measure. 12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

By the Vertical Angle Theorem, 10. BC

SOLUTION: The opposite sides of a rectangle are parallel and ANSWER: congruent. Therefore, BC = AD = 2 ft. 50

ANSWER: 13. 2 ft SOLUTION: The diagonals of a rectangle are congruent and 11. DB bisect each other. So, is an isosceles SOLUTION: triangle.

All four angles of a rectangle are right angles. So, Then, All four angles of a rectangle are right angles. is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6 ANSWER: 25 CCSS REGULARITY Quadrilateral WXYZ is a ANSWER: rectangle. 6.3 ft

12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles 14. If ZY = 2x + 3 and WX = x + 4, find WX. triangle. Then, SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 By the Vertical Angle Theorem, Use the value of x to find WX.

WX = 1 + 4 = 5. ANSWER: ANSWER: 50 5 13. 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, is an isosceles The diagonals of a rectangle bisect each other. So, triangle. PY = WP. Then, All four angles of a rectangle are right angles. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

ANSWER: 25 ANSWER: CCSS REGULARITY Quadrilateral WXYZ is a 43 rectangle. 16. If , find . SOLUTION: All four angles of a rectangle are right angles. So,

14. If ZY = 2x + 3 and WX = x + 4, find WX.

SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5. ANSWER: 39 ANSWER: 5 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY. SOLUTION: The diagonals of a rectangle bisect each other. So, PY = WP.

Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. ANSWER: 38

18. If , find . ANSWER: SOLUTION: 43 All four angles of a rectangle are right angles. So, 16. If , find . SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 48

19. If , find . SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 39 is a right triangle. Then, 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY. Therefore, Since is a transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, Then ZP = 4(7) – 9 = 19. = = 46. Therefore, ZX = 2(ZP) = 38. ANSWER: ANSWER: 46 38 PROOF Write a two-column proof. 18. If , find 20. Given: ABCD is a rectangle. . Prove: SOLUTION: All four angles of a rectangle are right angles. So,

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. ANSWER: You need to prove . Use the 48 properties that you have learned about rectangles to walk through the proof. 19. If , find . Given: ABCD is a rectangle. Prove: SOLUTION: Proof: All four angles of a rectangle are right angles. So, Statements (Reasons) is a right triangle. Then, 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) Therefore, Since is a 6. (SSS) transversal of parallel sides and , alternate ANSWER: interior angles WZX and ZXY are congruent. So, Proof: = = 46. Statements (Reasons) ANSWER: 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 46 3. (Opp. sides of a .) PROOF Write a two-column proof. 4. (Refl. Prop.) 20. Given: ABCD is a rectangle. 5. (Diagonals of a rectangle are .) Prove: 6. (SSS)

21. Given: QTVW is a rectangle.

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the Prove: properties that you have learned about rectangles to SOLUTION: walk through the proof. You need to walk through the proof step by step. Look over what you are given and what you need to Given: ABCD is a rectangle. prove. Here, you are given QTVW is a rectangle and Prove: Proof: . You need to prove . Use Statements (Reasons) the properties that you have learned about rectangles 1. ABCD is a rectangle. (Given) to walk through the proof. 2. ABCD is a parallelogram. (Def. of rectangle) Given: QTVW is a rectangle. 3. (Opp. sides of a .)

4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS)

ANSWER: Proof: Prove: Statements (Reasons) Proof: 1. ABCD is a rectangle. (Given) Statements (Reasons) 2. ABCD is a parallelogram. (Def. of rectangle) 1. QTVW is a rectangle; .(Given)

3. (Opp. sides of a .) 2. QTVW is a parallelogram. (Def. of rectangle)

4. (Refl. Prop.) 3. (Opp sides of a .) 5. (Diagonals of a rectangle are .) 4. are right angles.(Def. of rectangle) 6. (SSS) 5. (All rt .) 6. QR = ST (Def. of segs.) Given: 21. QTVW is a rectangle. 7. (Refl. Prop.)

8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.)

Prove: 13. (SAS) SOLUTION: ANSWER: You need to walk through the proof step by step. Proof: Look over what you are given and what you need to Statements (Reasons) prove. Here, you are given QTVW is a rectangle and 1. QTVW is a rectangle; .(Given) . You need to prove . Use 2. QTVW is a parallelogram. (Def. of rectangle) the properties that you have learned about rectangles 3. (Opp sides of a .) to walk through the proof. 4. are right angles.(Def. of rectangle)

Given: QTVW is a rectangle. 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) Prove: 12. (Def. of segs.) Proof:

Statements (Reasons) 13. (SAS)

1. QTVW is a rectangle; .(Given) COORDINATE GEOMETRY Graph each 2. QTVW is a parallelogram. (Def. of rectangle) quadrilateral with the given vertices. Determine 3. (Opp sides of a .) whether the figure is a rectangle. Justify your 4. are right angles.(Def. of rectangle) answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula 5. (All rt .) 6. QR = ST (Def. of segs.) SOLUTION: 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) Use the slope formula to find the slope of the sides of 1. QTVW is a rectangle; .(Given) the quadrilateral. 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. 12. (Def. of segs.) Therefore, the quadrilateral WXYZ is a parallelogram. 13. (SAS) The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to COORDINATE GEOMETRY Graph each each other. That is, all the four angles are right quadrilateral with the given vertices. Determine angles. Therefore, WXYZ is a rectangle. whether the figure is a rectangle. Justify your answer using the indicated formula. ANSWER: 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula Yes; slope of slope of , slope SOLUTION: of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

ANSWER:

Yes; slope of slope of , slope First, Use the Distance formula to find the lengths of of slope of . So WXYZ is a the sides of the quadrilateral. parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance No; so JKLM is a Formula parallelogram; so SOLUTION: the diagonals are not congruent. Thus, JKLM is not a rectangle.

First, Use the Distance formula to find the lengths of

the sides of the quadrilateral. 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula

SOLUTION:

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle. First, use the Distance Formula to find the lengths of the sides of the quadrilateral. ANSWER:

No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle. so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER:

Yes; so QRST is a parallelogram. so the diagonals are 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula congruent. QRST is a rectangle. SOLUTION:

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula First, use the Distance Formula to find the lengths of the sides of the quadrilateral. SOLUTION:

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle. First, use the slope formula to find the lengths of the sides of the quadrilateral. ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula ANSWER: SOLUTION: No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

First, use the slope formula to find the lengths of the sides of the quadrilateral.

Quadrilateral ABCD is a rectangle. Find each measure if .

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a 26. parallelogram. None of the adjacent sides have slopes whose SOLUTION: product is –1. So, the angles are not right angles. All four angles of a rectangle are right angles. So, Therefore, GHJK is not a rectangle. ANSWER:

No; slope of slope of and slope of ANSWER: slope of . So, GHJK is a 50 parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are 27. not perpendicular. Thus, GHJK is not a rectangle. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: 40

28. SOLUTION: Quadrilateral ABCD is a rectangle. Find each The measures of angles 2 and 3 are equal as they measure if . are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. 26. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they are alternate interior angles. ANSWER: Therefore, m∠4 = m∠1 = 50. 50 Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m 6 = m 4 = 50. 27. ∠ ∠ The sum of the three angles of a triangle is 180. SOLUTION: Therefore, m∠5 = 180 – (50 + 50) = 80. The measures of angles 2 and 3 are equal as they are alternate interior angles. ANSWER: Since the diagonals of a rectangle are congruent and 80 bisect each other, the triangle with the angles 3, 7

and 8 is an isosceles triangle. So, m∠7 = m∠3. 30. Therefore, m∠7 = m∠2 = 40. SOLUTION: All four angles of a rectangle are right angles. So, ANSWER:

28. The measures of angles 1 and 4 are equal as they SOLUTION: are alternate interior angles. The measures of angles 2 and 3 are equal as they Therefore, m∠4 = m∠1 = 50. are alternate interior angles. Since the diagonals of a rectangle are congruent and Therefore, m∠3 = m∠2 = 40. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m 6 = ANSWER: ∠ m∠4 = 50. 40 ANSWER: 29. 50 SOLUTION: All four angles of a rectangle are right angles. So, 31. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. The measures of angles 1 and 4 are equal as they Since the diagonals of a rectangle are congruent and are alternate interior angles. bisect each other, the triangle with the angles 3, 7 Therefore, m∠4 = m∠1 = 50. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. Since the diagonals of a rectangle are congruent and The sum of the three angles of a triangle is 180. bisect each other, the triangle with the angles 4, 5 Therefore, m∠8 = 180 – (40 + 40) = 100. and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. ANSWER: Therefore, m∠5 = 180 – (50 + 50) = 80. 100 ANSWER: 32. CCSS MODELING Jody is building a new 80 bookshelf using wood and metal supports like the one shown. To what length should she cut the metal 30. supports in order for the bookshelf to be square, which means that the angles formed by the shelves SOLUTION: and the vertical supports are all right angles? Explain All four angles of a rectangle are right angles. So, your reasoning.

m∠4 = 50. In order for the bookshelf to be square, the lengths of ANSWER: all of the metal supports should be equal. Since we know the length of the bookshelves and the distance 50 between them, use the Pythagorean Theorem to find the lengths of the metal supports. 31. Let x be the length of each support. Express 3 feet SOLUTION: as 36 inches. The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. Therefore, the metal supports should each be 39 Therefore, m∠8 = 180 – (40 + 40) = 100. inches long or 3 feet 3 inches long. ANSWER: ANSWER: 100 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports 32. CCSS MODELING Jody is building a new should be equal. Since I knew the length of the bookshelf using wood and metal supports like the one bookshelves and the distance between them, I used shown. To what length should she cut the metal the Pythagorean Theorem to find the lengths of the supports in order for the bookshelf to be square, metal supports. which means that the angles formed by the shelves and the vertical supports are all right angles? Explain PROOF Write a two-column proof. your reasoning. 33. Theorem 6.13 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

SOLUTION: Given: WXYZ is a rectangle with diagonals In order for the bookshelf to be square, the lengths of . all of the metal supports should be equal. Since we know the length of the bookshelves and the distance Prove: between them, use the Pythagorean Theorem to find the lengths of the metal supports. Let x be the length of each support. Express 3 feet as 36 inches.

Proof: 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) Therefore, the metal supports should each be 39

inches long or 3 feet 3 inches long. 3. (Refl. Prop.) 4. are right angles. (Def. of ANSWER: rectangle) 3 ft 3 in.; Sample answer: In order for the bookshelf 5. (All right .) to be square, the lengths of all of the metal supports 6. (SAS) should be equal. Since I knew the length of the bookshelves and the distance between them, I used 7. (CPCTC) the Pythagorean Theorem to find the lengths of the metal supports. ANSWER: Given: WXYZ is a rectangle with diagonals PROOF Write a two-column proof. . 33. Theorem 6.13 Prove: SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove . Use the properties that you have learned Proof: about rectangles to walk through the proof. 1. WXYZ is a rectangle with diagonals . (Given) Given: WXYZ is a rectangle with diagonals 2. (Opp. sides of a .)

. 3. (Refl. Prop.) Prove: 4. are right angles. (Def. of rectangle) 5. (All right .) 6. (SAS)

7. (CPCTC) Proof: 1. WXYZ is a rectangle with diagonals . 34. Theorem 6.14 (Given) SOLUTION:

2. (Opp. sides of a .) You need to walk through the proof step by step. 3. (Refl. Prop.) Look over what you are given and what you need to 4. are right angles. (Def. of prove. Here, you are given rectangle) . You need to 5. (All right .) prove that WXYZ is a rectangle.. Use the properties 6. (SAS) that you have learned about rectangles to walk

through the proof. 7. (CPCTC)

ANSWER: Given: Given: WXYZ is a rectangle with diagonals Prove: WXYZ is a rectangle. . Prove:

Proof: 1. (Given)

Proof: 2. (SSS) 1. WXYZ is a rectangle with diagonals . 3. (CPCTC) (Given) 4. (Def. of ) 2. (Opp. sides of a .) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 3. (Refl. Prop.) 6. are supplementary. (Cons. 4. are right angles. (Def. of are suppl.) rectangle) 7. (Def of suppl.) 5. (All right .) 8. are right angles. (If 6. (SAS) 2 and suppl., each is a rt. .) 7. (CPCTC) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 34. Theorem 6.14 10. WXYZ is a rectangle. (Def. of rectangle) SOLUTION: ANSWER: You need to walk through the proof step by step.

Look over what you are given and what you need to Given: prove. Here, you are given Prove: WXYZ is a rectangle. . You need to prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk through the proof.

Given: Proof: Prove: WXYZ is a rectangle. 1. (Given) 2. (SSS) 3. (CPCTC) 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .)

Proof: 6. are supplementary. (Cons.

1. (Given) are suppl.) 2. (SSS) 7. (Def of suppl.) 3. (CPCTC) 8. are right angles. (If 4. (Def. of ) 2 and suppl., each is a rt. .) 5. WXYZ is a parallelogram. (If both pairs of opp. 9. are right angles. (If a has 1 sides are , then quad. is .) rt. , it has 4 rt. .) 6. are supplementary. (Cons. 10. WXYZ is a rectangle. (Def. of rectangle) are suppl.) 7. (Def of suppl.) PROOF Write a paragraph proof of each 8. are right angles. (If statement. 35. If a parallelogram has one right angle, then it is a 2 and suppl., each is a rt. .) rectangle. 9. are right angles. (If a has 1 SOLUTION: rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) You need to walk through the proof step by step. Look over what you are given and what you need to ANSWER: prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. Given: Let ABCD be a parallelogram with one right angle. Prove: WXYZ is a rectangle. Then use the properties of parallelograms to walk through the proof.

Proof: 1. (Given)

2. (SSS) ABCD is a parallelogram, and is a right angle. 3. (CPCTC) Since ABCD is a parallelogram and has one right 4. (Def. of ) angle, then it has four right angles. So by the 5. WXYZ is a parallelogram. (If both pairs of opp. definition of a rectangle, ABCD is a rectangle. sides are , then quad. is .) ANSWER: 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 ABCD is a parallelogram, and is a right angle. rt. , it has 4 rt. .) Since ABCD is a parallelogram and has one right 10. WXYZ is a rectangle. (Def. of rectangle) angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. PROOF Write a paragraph proof of each statement. 35. If a parallelogram has one right angle, then it is a 36. If a quadrilateral has four right angles, then it is a rectangle. rectangle. SOLUTION: SOLUTION: You need to walk through the proof step by step. You need to walk through the proof step by step. Look over what you are given and what you need to Look over what you are given and what you need to prove. Here, you are given a parallelogram with one prove. Here, you are given a quadrilateral with four right angle. You need to prove that it is a rectangle. right angles. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms to walk Then use the properties of parallelograms and through the proof. rectangles to walk through the proof.

ANSWER: ANSWER:

ABCD is a parallelogram, and is a right angle. ABCD is a quadrilateral with four right angles. Since ABCD is a parallelogram and has one right ABCD is a parallelogram because both pairs of angle, then it has four right angles. So by the opposite angles are congruent. By definition of a definition of a rectangle, ABCD is a rectangle. rectangle, ABCD is a rectangle. 37. CONSTRUCTION Construct a rectangle using the 36. If a quadrilateral has four right angles, then it is a construction for congruent segments and the rectangle. construction for a line perpendicular to another line SOLUTION: through a point on the line. Justify each step of the

You need to walk through the proof step by step. construction. Look over what you are given and what you need to SOLUTION: prove. Here, you are given a quadrilateral with four Sample answer: right angles. You need to prove that it is a rectangle. Step 1: Graph a line and place points P and Q on the Let ABCD be a quadrilateral with four right angles. line. Then use the properties of parallelograms and rectangles to walk through the proof. Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and right of P that intersects .

Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Place the compass on each point of intersection and ABCD is a quadrilateral with four right angles. draw arcs that intersect above . ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3. ANSWER: Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass ABCD is a quadrilateral with four right angles. at P and draw an arc above it that intersects line a. ABCD is a parallelogram because both pairs of Label the point of intersection R. opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that 37. CONSTRUCTION Construct a rectangle using the intersects line b. Label the point of intersection S. construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the Step 9: Draw . construction. SOLUTION: Sample answer: Step 1: Graph a line and place points P and Q on the line.

Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and right of P that intersects .

Step 3: Open the compass to a setting greater than Since and , the measure of the distance from P to either point of intersection. angles P and Q is 90 degrees. Lines that are Place the compass on each point of intersection and perpendicular to the same line are parallel, so draw arcs that intersect above . . The same compass setting was used to

locate points R and S, so . If one pair of Step 4: Draw line a through point P and the opposite sides of a quadrilateral is both parallel and

intersection of the arcs drawn in step 3. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Step 5: Repeat steps 2 and 3 using point Q. Thus, PRSQ is a rectangle.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5. ANSWER:

Step 7: Using any compass setting, put the compass Sample answer: Since and , the at P and draw an arc above it that intersects line a. measure of angles P and Q is 90 degrees. Lines that Label the point of intersection R. are perpendicular to the same line are parallel, so

Step 8: Using the same compass setting, put the . The same compass setting was used to compass at Q and draw an arc above it that locate points R and S, so . If one pair of intersects line b. Label the point of intersection S. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 9: Draw . A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

Since and , the measure of angles P and Q is 90 degrees. Lines that are 38. SPORTS The end zone of a football field is 160 feet perpendicular to the same line are parallel, so wide and 30 feet long. Kyle is responsible for . The same compass setting was used to painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is locate points R and S, so . If one pair of the regulation size and be sure that it is also a opposite sides of a quadrilateral is both parallel and rectangle using only a tape measure. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. SOLUTION: Thus, PRSQ is a rectangle. Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both ANSWER: pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then Sample answer: Since and , the the end zone is a rectangle. measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so ANSWER: . The same compass setting was used to Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the locate points R and S, so . If one pair of diagonals and both pairs of opposite sides are opposite sides of a quadrilateral is both parallel and congruent, then the end zone is a rectangle. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. ALGEBRA Quadrilateral WXYZ is a rectangle. Thus, PRSQ is a rectangle.

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right 2 2 triangle. By the Pythagorean Theorem, XZ = XW + 2 WZ . 38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is Therefore, YW = 5. the regulation size and be sure that it is also a rectangle using only a tape measure. ANSWER: 5 SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. measure the diagonals of the end zone and both SOLUTION: pairs of opposite sides. If both pairs of opposite sides The diagonals of a rectangle are congruent to each are congruent, then the quadrilateral is a other. So, WY = XZ = 2c. All four angles of a parallelogram. If the diagonals are congruent, then the end zone is a rectangle. rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XY2 + ANSWER: 2 ZY . Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle. Therefore, WY = 2(5) = 10.

ANSWER: 10

39. If XW = 3, WZ = 4, and XZ = b, find YW. 41. SIGNS The sign below is in the foyer of Nyoko’s SOLUTION: school. Based on the dimensions given, can Nyoko The diagonals of a rectangle are congruent to each be sure that the sign is a rectangle? Explain your other. So, YW = XZ = b. All four angles of a reasoning. rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 WZ .

Therefore, YW = 5. ANSWER: SOLUTION: 5 Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. can be used to prove that it is a rectangle. If the SOLUTION: diagonals could be proven to be congruent, or if the The diagonals of a rectangle are congruent to each angles could be proven to be right angles, then we

other. So, WY = XZ = 2c. All four angles of a could prove that the sign was a rectangle. rectangle are right angles. So, is a right ANSWER: 2 2 triangle. By the Pythagorean Theorem, XZ = XY + No; sample answer: Both pairs of opposite sides are 2 ZY . congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent. Therefore, WY = 2(5) = 10. SOLUTION: ANSWER: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let 10 the length of the bases be a units and the height be b 41. SIGNS The sign below is in the foyer of Nyoko’s units. Then the rest of the vertices are A(0, b), B(a, school. Based on the dimensions given, can Nyoko b), and C(a, 0). You need to walk through the proof be sure that the sign is a rectangle? Explain your step by step. Look over what you are given and what reasoning. you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove:

SOLUTION:

Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the

diagonals could be proven to be congruent, or if the Proof: angles could be proven to be right angles, then we Use the Distance Formula to find the lengths of the

could prove that the sign was a rectangle. diagonals. ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. The diagonals, have the same length, so they are congruent. PROOF Write a coordinate proof of each statement. ANSWER: 42. The diagonals of a rectangle are congruent. Given: ABCD is a rectangle. Prove: SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the Proof: properties that you have learned about rectangles to Use the Distance Formula to find

walk through the proof. and .

Given: ABCD is a rectangle. have the same length, so they are

Prove: congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c Proof: units. Then the rest of the vertices are B(a, 0), C(b + Use the Distance Formula to find the lengths of the a, c), and D(b, c). You need to walk through the diagonals. proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the The diagonals, have the same length, so proof. they are congruent. Given: ABCD and ANSWER:

Given: ABCD is a rectangle. Prove: ABCD is a rectangle. Prove:

Proof:

Proof: Use the Distance Formula to find and . AC = BD. So, have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a Because A and B are different points, coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c a ≠ 0. Then b = 0. The slope of is undefined units. Then the rest of the vertices are B(a, 0), C(b + and the slope of . Thus, . ∠DAB is a, c), and D(b, c). You need to walk through the a right angle and ABCD is a rectangle. proof step by step. Look over what you are given and what you need to prove. Here, you are given ANSWER: ABCD and and you need to prove that Given: ABCD and ABCD is a rectangle. Use the properties that you Prove: ABCD is a rectangle. have learned about rectangles to walk through the proof.

Given: ABCD and Prove: ABCD is a rectangle.

Proof:

But AC = BD. So, Proof:

AC = BD. So,

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

Because A and B are different points, 44. MULTIPLE REPRESENTATIONS In the a ≠ 0. Then b = 0. The slope of is undefined problem, you will explore properties of other special parallelograms. and the slope of . Thus, . ∠DAB is a. GEOMETRIC Draw three parallelograms, each a right angle and ABCD is a rectangle. with all four sides congruent. Label one ANSWER: parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and

Given: ABCD and label the intersections R. Prove: ABCD is a rectangle. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: Proof: a. Sample answer:

But AC = BD. So,

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined b. Use a protractor to measure each angle listed in the table. and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the c. Sample answer: Each of the angles listed in the problem, you will explore properties of other special table are right angles. These angles were each parallelograms. formed by diagonals. The diagonals of a a. GEOMETRIC Draw three parallelograms, each parallelogram with four congruent sides are with all four sides congruent. Label one perpendicular. parallelogram ABCD, one MNOP, and one WXYZ. ANSWER: Draw the two diagonals of each parallelogram and label the intersections R. a. Sample answer: b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y. b. Use a protractor to measure each angle listed in the table.

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular. ANSWER: SOLUTION: a. Sample answer: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So,

The sum of the angles of a triangle is 180. b. So, By the Vertical Angle Theorem,

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the ANSWER: values of x and y. x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a SOLUTION: single vertex of a triangle. All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and

bisect each other. So, is an isosceles triangle In order for the quadrilateral to be a rectangle, one of with the angles in the congruent triangles has to be a right So, angle.

The sum of the angles of a triangle is 180. So,

By the Vertical Angle Theorem, So, Tamika is correct.

46. CCSS CRITIQUE Parker says that any two 47. REASONING In the diagram at the right, lines n, p, congruent acute triangles can be arranged to make a q,and r are parallel and lines l and m are parallel. rectangle. Tamika says that only two congruent right How many rectangles are formed by the intersecting triangles can be arranged to make a rectangle. Is lines? either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the In order for the quadrilateral to be a rectangle, one of lines intersect the line m at the points W, X, Y, and Z. the angles in the congruent triangles has to be a right angle.

So, Tamika is correct.

ANSWER: Tamika; Sample answer: When two congruent Then the rectangles formed are ABXW, ACYW, triangles are arranged to form a quadrilateral, two of BCYX, BDZX, CDZY, and ADZW. Therefore, 6 the angles are formed by a single vertex of a triangle. rectangles are formed by the intersecting lines. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right ANSWER: angle. 6

47. REASONING In the diagram at the right, lines n, p, 48. OPEN ENDED Write the equations of four lines q,and r are parallel and lines l and m are parallel. having intersections that form the vertices of a How many rectangles are formed by the intersecting rectangle. Verify your answer using coordinate lines? geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 SOLUTION: sides. Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the x = 0, x = 6, y = 0, y = 4; lines intersect the line m at the points W, X, Y, and Z. eSolutions Manual - Powered by Cognero Page 18

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the Then the rectangles formed are ABXW, ACYW, quadrilateral is both parallel and congruent, by BCYX, BDZX, CDZY, and ADZW. Therefore, 6 Theorem 6.12, the quadrilateral is a parallelogram. rectangles are formed by the intersecting lines. Since is horizontal and is vertical, the lines ANSWER: are perpendicular and the measure of the angle they 6 form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by 48. OPEN ENDED Write the equations of four lines definition, the parallelogram is a rectangle. having intersections that form the vertices of a ANSWER: rectangle. Verify your answer using coordinate geometry. Sample answer: x = 0, x = 6, y = 0, y = 4; SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

x = 0, x = 6, y = 0, y = 4; The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. The length of is 6 − 0 or 6 units and the length of 49. WRITING IN MATH Explain why all rectangles is 6 – 0 or 6 units. The slope of AB is 0 and the are parallelograms, but all parallelograms are not slope of DC is 0. Since a pair of sides of the rectangles. quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. SOLUTION: Since is horizontal and is vertical, the lines Sample answer: A parallelogram is a quadrilateral in are perpendicular and the measure of the angle they which each pair of opposite sides are parallel. By form is 90. By Theorem 6.6, if a parallelogram has definition of rectangle, all rectangles have both pairs one right angle, it has four right angles. Therefore, by of opposite sides parallel. So rectangles are always

definition, the parallelogram is a rectangle. parallelograms.

ANSWER: Parallelograms with right angles are rectangles, so Sample answer: x = 0, x = 6, y = 0, some parallelograms are rectangles, but others with non-right angles are not. y = 4; ANSWER: Sample answer: All rectangles are parallelograms because, by definition, both pairs of opposite sides are parallel. Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram FGHJ a rectangle? The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines A x = 3, y = 4 are perpendicular and the measure of the angle they B x = 4, y = 3 form is 90. By Theorem 6.6, if a parallelogram has C x = 7, y = 8 one right angle, it has four right angles. Therefore, by D x = 8, y = 7 definition, the parallelogram is a rectangle. SOLUTION: 49. WRITING IN MATH Explain why all rectangles The opposite sides of a parallelogram are congruent. are parallelograms, but all parallelograms are not So, FJ = GH. rectangles. –3x + 5y = 11 FJ = GH

SOLUTION: If the diagonals of a parallelogram are congruent, Sample answer: A parallelogram is a quadrilateral in then it is a rectangle. Since FGHJ is a parallelogram, which each pair of opposite sides are parallel. By the diagonals bisect each other. So, if it is a definition of rectangle, all rectangles have both pairs rectangle, then will be an isosceles triangle of opposite sides parallel. So rectangles are always with That is, 3x + y = 13. parallelograms.

Parallelograms with right angles are rectangles, so Add the two equations to eliminate the x-term and some parallelograms are rectangles, but others with solve for y. non-right angles are not. –3x + 5y = 11 3x + y = 13 ANSWER: 6y = 24 Add to eliminate x-term. Sample answer: All rectangles are parallelograms y = 4 Divide each side by 6. because, by definition, both pairs of opposite sides are parallel. Parallelograms with right angles are Use the value of y to find the value of x. rectangles, so some parallelograms are rectangles, 3x + 4 = 13 Substitute. but others with non-right angles are not. 3x = 9 Subtract 4 from each side. x = 3 Divide each side by 3. 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram So, x = 3, y = 4, the correct choice is A. FGHJ a rectangle? ANSWER: A

51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 A x = 3, y = 4 feet longer than the other. Which of the following B x = 4, y = 3 equations could be used to find r, the shorter side of C x = 7, y = 8 the playground? D x = 8, y = 7 F 10r + r = 80 G 4r + 10 = 80 SOLUTION: H r(r + 10) = 80 The opposite sides of a parallelogram are congruent. J 2(r + 10) + 2r = 80 So, FJ = GH. –3x + 5y = 11 FJ = GH SOLUTION: The formula to find the perimeter of a rectangle is If the diagonals of a parallelogram are congruent, given by P = 2(l + w). The perimeter is given to be 8 then it is a rectangle. Since FGHJ is a parallelogram, ft. The width is r and the length is r + 10. So, 2r + 2 the diagonals bisect each other. So, if it is a (r + 10) = 80. rectangle, then will be an isosceles triangle Therefore, the correct choice is J. with That is, 3x + y = 13. ANSWER: Add the two equations to eliminate the x-term and J solve for y. SHORT RESPONSE –3x + 5y = 11 52. What is the measure of 3x + y = 13 ? 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6.

Use the value of y to find the value of x. 3x + 4 = 13 Substitute. 3x = 9 Subtract 4 from each side. x = 3 Divide each side by 3. SOLUTION:

The diagonals of a rectangle are congruent and So, x = 3, y = 4, the correct choice is A. bisect each other. So, is an isosceles triangle ANSWER: with A So, By the Exterior Angle Theorem, 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following ANSWER: equations could be used to find r, the shorter side of 112 the playground? F 10r + r = 80 53. SAT/ACT If p is odd, which of the following must G 4r + 10 = 80 also be odd? H r(r + 10) = 80 A 2p J 2(r + 10) + 2r = 80 B 2p + 2

SOLUTION: C The formula to find the perimeter of a rectangle is D given by P = 2(l + w). The perimeter is given to be 8 E p + 2 ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. SOLUTION: Therefore, the correct choice is J. Analyze each answer choice to determine which expression is odd when p is odd. ANSWER: Choice A 2p J This is a multiple of 2 so it is even. Choice B 2p + 2 52. SHORT RESPONSE What is the measure of This can be rewritten as 2(p + 1) which is also a ? multiple of 2 so it is even.

Choice C If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. SOLUTION: Choice E p + 2 The diagonals of a rectangle are congruent and Adding 2 to an odd number results in an odd bisect each other. So, is an isosceles triangle number. This is odd. with The correct choice is E. So, By the Exterior Angle Theorem, ANSWER: E

ANSWER: ALGEBRA Find x and y so that the 112 quadrilateral is a parallelogram.

53. SAT/ACT If p is odd, which of the following must also be odd? A 2p B 2p + 2 54.

C SOLUTION: D The opposite sides of a parallelogram are congruent. E p + 2

SOLUTION: Analyze each answer choice to determine which In a parallelogram, consecutive interior angles are expression is odd when p is odd. supplementary. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

Choice C ANSWER: If p is odd , is not a whole number. x = 2, y = 41 Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. The correct choice is E. 55. ANSWER: SOLUTION: E The opposite sides of a parallelogram are congruent.

ALGEBRA Find x and y so that the quadrilateral is a parallelogram. The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal. 54. SOLUTION: The opposite sides of a parallelogram are congruent.

ANSWER: x = 8, y = 22 In a parallelogram, consecutive interior angles are supplementary.

56. SOLUTION: The diagonals of a parallelogram bisect each other. ANSWER: So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. x = 2, y = 41 Solve the first equation to find the value of y. 2y = 14 y = 7 Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x 2 = x 55. ANSWER: SOLUTION:

The opposite sides of a parallelogram are congruent. x = 2, y = 7 57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C The opposite sides of a parallelogram are parallel to (4, –2), and D(–1, –1). each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and SOLUTION: hence they are equal. The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or .

ANSWER: Therefore, the coordinates of the intersection of the x = 8, y = 22 diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure. 56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 Use the value of y in the second equation and solve

for x. 58. If , name two congruent angles. 7 + 3 = 2x + 6 SOLUTION: 4 = 2x 2 = x Since is an isosceles triangle. So,

ANSWER: x = 2, y = 7 ANSWER:

57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C 59. If , name two congruent segments. (4, –2), and D(–1, –1). SOLUTION: SOLUTION: In a triangle with two congruent angles, the sides The diagonals of a parallelogram bisect each other, opposite to the congruent angles are also congruent. so the intersection is the midpoint of either diagonal. So, . Find the midpoint of or . ANSWER:

Therefore, the coordinates of the intersection of the 60. If , name two congruent segments. diagonals are (2.5, 0.5). SOLUTION: ANSWER: In a triangle with two congruent angles, the sides (2.5, 0.5) opposite to the congruent angles are also congruent. So, . Refer to the figure. ANSWER:

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER:

3. FARMING An X-brace on a rectangular barn SOLUTION: door is both decorative and functional. It helps to prevent the door from warping over time. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. If feet, PS = 7 feet, and Then, , find each measure. By the Exterior Angle Theorem,

Therefore,

ANSWER: 33.5

4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 1. QR Then, SOLUTION: By the Vertical Angle Theorem, The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft. Therefore, ANSWER: 7 ft ANSWER: 56.5 2. SQ ALGEBRA Quadrilateral DEFG is a rectangle. SOLUTION: The diagonals of a rectangle bisect each other. So,

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

ANSWER:

Use the value of x to find EG. EG = 6 + 5 = 11 3. ANSWER: SOLUTION: 11 The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 6. If , Then, find . By the Exterior Angle Theorem, SOLUTION: All four angles of a rectangle are right angles. So, Therefore,

ANSWER: 33.5

4. SOLUTION: The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 51 Then, By the Vertical Angle Theorem, 7. PROOF If ABDE is a rectangle and , prove that . Therefore,

ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. You are given ABDE is a rectangle and 5. If FD = 3x – 7 and EG = x + 5, find EG. . You need to prove . Use the SOLUTION: properties that you have learned about rectangles to The diagonals of a rectangle are congruent to each walk through the proof. other. So, FD = EG. Given: ABDE is a rectangle; Prove:

Use the value of x to find EG. EG = 6 + 5 = 11

ANSWER: 11

6. If , Statements(Reasons)

find . 1. ABDE is a rectangle; . (Given) SOLUTION: 2. ABDE is a parallelogram. (Def. of All four angles of a rectangle are right angles. So, rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS)

7. (CPCTC)

ANSWER: ANSWER: Given: ABDE is a rectangle; 51 Prove: 7. PROOF If ABDE is a rectangle and , prove that .

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of SOLUTION: rectangle) You need to walk through the proof step by step. 3. (Opp. sides of a .) Look over what you are given and what you need to prove. You are given ABDE is a rectangle and 4. are right angles.(Def. of rectangle) . You need to prove . Use the 5. (All rt .) properties that you have learned about rectangles to 6. (SAS) walk through the proof. 7. (CPCTC)

Given: ABDE is a rectangle; COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine Prove: whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) Use the slope formula to find the slope of the sides of 6. (SAS) the quadrilateral. 7. (CPCTC) ANSWER: Given: ABDE is a rectangle; Prove:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

Statements(Reasons) ANSWER: 1. ABDE is a rectangle; . No; slope of , slope of , slope of (Given) 2. ABDE is a parallelogram. (Def. of rectangle) , and slope of . Slope of

3. (Opp. sides of a .) slope of , and slope of slope of

4. are right angles.(Def. of rectangle) , so WXYZ is not a parallelogram. Therefore, 5. (All rt .) WXYZ is not a rectangle. 6. (SAS) 7. (CPCTC)

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. so ABCD is a ANSWER: parallelogram. A parallelogram is a rectangle if the diagonals are No; slope of , slope of , slope of congruent. Use the Distance formula to find the lengths of the diagonals. , and slope of . Slope of slope of , and slope of slope of So the diagonals are congruent. Thus, ABCD is a , so WXYZ is not a parallelogram. Therefore, rectangle. WXYZ is not a rectangle. ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral. 10. BC SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. so ABCD is a parallelogram. ANSWER: A parallelogram is a rectangle if the diagonals are 2 ft congruent. Use the Distance formula to find the lengths of the diagonals. 11. DB SOLUTION: So the diagonals are congruent. Thus, ABCD is a All four angles of a rectangle are right angles. So, rectangle. is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . ANSWER: BC = AD = 2 Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a DC = AB = 6 parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: 6.3 ft

12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure. By the Vertical Angle Theorem,

ANSWER: 50

13. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles 10. BC triangle. SOLUTION: Then, All four angles of a rectangle are right angles. The opposite sides of a rectangle are parallel and

congruent. Therefore, BC = AD = 2 ft.

ANSWER: 2 ft ANSWER: 25 11. DB SOLUTION: CCSS REGULARITY Quadrilateral WXYZ is a All four angles of a rectangle are right angles. So, rectangle. is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: ANSWER: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 6.3 ft 2x + 3 = x + 4 12. x = 1 Use the value of x to find WX. SOLUTION: WX = 1 + 4 = 5. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles ANSWER: triangle. 5 Then, 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: The diagonals of a rectangle bisect each other. So, By the Vertical Angle Theorem, PY = WP.

ANSWER: 50 The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 13. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles ANSWER: triangle. Then, 43 All four angles of a rectangle are right angles. 16. If , find

. SOLUTION: All four angles of a rectangle are right angles. So, ANSWER: 25 CCSS REGULARITY Quadrilateral WXYZ is a rectangle.

ANSWER: 39

14. If ZY = 2x + 3 and WX = x + 4, find WX. 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: SOLUTION: The opposite sides of a rectangle are parallel and The diagonals of a rectangle are congruent and congruent. So, ZY = WX. bisect each other. So, ZP = PY. 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5. Then ZP = 4(7) – 9 = 19. ANSWER: Therefore, ZX = 2(ZP) = 38. 5 ANSWER: 38 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: 18. If , find The diagonals of a rectangle bisect each other. So, . PY = WP. SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle.

ANSWER: ANSWER: 43 48

16. If , find 19. If , find

. . SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, All four angles of a rectangle are right angles. So, is a right triangle. Then,

Therefore, Since is a transversal of parallel sides and , alternate ANSWER: interior angles WZX and ZXY are congruent. So, 39 = = 46.

17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. ANSWER: SOLUTION: 46 The diagonals of a rectangle are congruent and PROOF Write a two-column proof.

bisect each other. So, ZP = PY. 20. Given: ABCD is a rectangle. Prove:

Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38.

ANSWER: SOLUTION: 38 You need to walk through the proof step by step. 18. If , find Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. . You need to prove . Use the SOLUTION: properties that you have learned about rectangles to All four angles of a rectangle are right angles. So, walk through the proof.

Given: ABCD is a rectangle. Prove: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) ANSWER: 3. (Opp. sides of a .) 48 4. (Refl. Prop.) 19. If , find 5. (Diagonals of a rectangle are .) . 6. (SSS) SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, Proof: is a right triangle. Then, Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.)

Therefore, Since is a 5. (Diagonals of a rectangle are .)

transversal of parallel sides and , alternate 6. (SSS) interior angles WZX and ZXY are congruent. So, = = 46. 21. Given: QTVW is a rectangle.

ANSWER: 46 PROOF Write a two-column proof. Given: 20. ABCD is a rectangle. Prove: Prove: SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and SOLUTION: . You need to prove . Use You need to walk through the proof step by step. the properties that you have learned about rectangles Look over what you are given and what you need to to walk through the proof. prove. Here, you are given ABCD is a rectangle. You need to prove . Use the Given: QTVW is a rectangle. properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove: Proof: Statements (Reasons) Prove: 1. ABCD is a rectangle. (Given) Proof: 2. ABCD is a parallelogram. (Def. of rectangle) Statements (Reasons) 3. (Opp. sides of a .) 1. QTVW is a rectangle; .(Given) 4. (Refl. Prop.) 2. QTVW is a parallelogram. (Def. of rectangle)

5. (Diagonals of a rectangle are .) 3. (Opp sides of a .) 6. (SSS) 4. are right angles.(Def. of rectangle) 5. (All rt .) ANSWER: 6. QR = ST (Def. of segs.) Proof: Statements (Reasons) 7. (Refl. Prop.) 1. ABCD is a rectangle. (Given) 8. RS = RS (Def. of segs.) 2. ABCD is a parallelogram. (Def. of rectangle) 9. QR + RS = RS + ST (Add. prop.) 3. (Opp. sides of a .) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 4. (Refl. Prop.) 12. (Def. of segs.) 5. (Diagonals of a rectangle are .) 13. (SAS) 6. (SSS) ANSWER: 21. Given: QTVW is a rectangle. Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle)

Prove: 5. (All rt .) 6. QR = ST (Def. of segs.) SOLUTION: 7. (Refl. Prop.) You need to walk through the proof step by step. Look over what you are given and what you need to 8. RS = RS (Def. of segs.) prove. Here, you are given QTVW is a rectangle and 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) . You need to prove . Use 11. QS = RT (Subst.) the properties that you have learned about rectangles to walk through the proof. 12. (Def. of segs.) 13. (SAS) Given: QTVW is a rectangle. COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION: Prove: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.)

7. (Refl. Prop.) 8. RS = RS (Def. of segs.) Use the slope formula to find the slope of the sides of 9. QR + RS = RS + ST (Add. prop.) the quadrilateral. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) The slopes of each pair of opposite sides are equal. 3. (Opp sides of a .) So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. 4. are right angles.(Def. of rectangle) The products of the slopes of the adjacent sides are 5. (All rt .) –1. So, any two adjacent sides are perpendicular to 6. QR = ST (Def. of segs.) each other. That is, all the four angles are right

7. (Refl. Prop.) angles. Therefore, WXYZ is a rectangle. 8. RS = RS (Def. of segs.) ANSWER: 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) Yes; slope of slope of , slope 11. QS = RT (Subst.) 12. (Def. of segs.) of slope of . So WXYZ is a parallelogram. The product of the slopes of 13. (SAS) consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a COORDINATE GEOMETRY Graph each rectangle. quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula SOLUTION:

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. First, Use the Distance formula to find the lengths of Therefore, the quadrilateral WXYZ is a parallelogram. the sides of the quadrilateral. The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle.

ANSWER: so JKLM is a parallelogram. Yes; slope of slope of , slope A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the of slope of . So WXYZ is a lengths of the diagonals. parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle. So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

First, use the Distance Formula to find the lengths of so JKLM is a the sides of the quadrilateral. parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

so QRST is a parallelogram. So the diagonals are not congruent. Thus, JKLM is A parallelogram is a rectangle if the diagonals are not a rectangle. congruent. Use the Distance Formula to find the lengths of the diagonals. ANSWER: No; so JKLM is a parallelogram; so So the diagonals are congruent. Thus, QRST is a the diagonals are not congruent. Thus, JKLM is not a rectangle. rectangle. ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

First, use the slope formula to find the lengths of the sides of the quadrilateral. so QRST is a

parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Yes; so QRST is a Therefore, the quadrilateral GHJK is a parallelogram. so the diagonals are parallelogram. congruent. QRST is a rectangle. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle.

ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

Quadrilateral ABCD is a rectangle. Find each measure if .

First, use the slope formula to find the lengths of the sides of the quadrilateral.

26. SOLUTION: All four angles of a rectangle are right angles. So,

The slopes of each pair of opposite sides are equal. ANSWER: So, the two pairs of opposite sides are parallel. 50 Therefore, the quadrilateral GHJK is a parallelogram. None of the adjacent sides have slopes whose 27. product is –1. So, the angles are not right angles. SOLUTION: Therefore, GHJK is not a rectangle. The measures of angles 2 and 3 are equal as they ANSWER: are alternate interior angles. Since the diagonals of a rectangle are congruent and No; slope of slope of and slope of bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. slope of . So, GHJK is a Therefore, m∠7 = m∠2 = 40. parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are ANSWER: not perpendicular. Thus, GHJK is not a rectangle. 40

28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER: 40

29. SOLUTION: All four angles of a rectangle are right angles. So, Quadrilateral ABCD is a rectangle. Find each measure if .

The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and 26. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. SOLUTION: The sum of the three angles of a triangle is 180. All four angles of a rectangle are right angles. So, Therefore, m∠5 = 180 – (50 + 50) = 80.

ANSWER: 80

ANSWER: 30. 50 SOLUTION: All four angles of a rectangle are right angles. So, 27. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. The measures of angles 1 and 4 are equal as they Since the diagonals of a rectangle are congruent and are alternate interior angles. bisect each other, the triangle with the angles 3, 7 Therefore, m∠4 = m∠1 = 50. and 8 is an isosceles triangle. So, m∠7 = m∠3. Since the diagonals of a rectangle are congruent and Therefore, m∠7 = m∠2 = 40. bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = ANSWER: m∠4 = 50. 40 ANSWER: 28. 50

SOLUTION: 31. The measures of angles 2 and 3 are equal as they are alternate interior angles. SOLUTION: Therefore, m∠3 = m∠2 = 40. The measures of angles 2 and 3 are equal as they are alternate interior angles. ANSWER: Since the diagonals of a rectangle are congruent and 40 bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. 29. The sum of the three angles of a triangle is 180.

SOLUTION: Therefore, m∠8 = 180 – (40 + 40) = 100. All four angles of a rectangle are right angles. So, ANSWER: 100

32. CCSS MODELING Jody is building a new The measures of angles 1 and 4 are equal as they bookshelf using wood and metal supports like the one are alternate interior angles. shown. To what length should she cut the metal Therefore, m∠4 = m∠1 = 50. supports in order for the bookshelf to be square, Since the diagonals of a rectangle are congruent and which means that the angles formed by the shelves bisect each other, the triangle with the angles 4, 5 and the vertical supports are all right angles? Explain your reasoning. and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: 80

30. SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find The measures of angles 1 and 4 are equal as they the lengths of the metal supports. are alternate interior angles. Let x be the length of each support. Express 3 feet

Therefore, m∠4 = m∠1 = 50. as 36 inches. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = m∠4 = 50. ANSWER: 50 Therefore, the metal supports should each be 39 inches long or 3 feet 3 inches long. 31. ANSWER: SOLUTION: 3 ft 3 in.; Sample answer: In order for the bookshelf The measures of angles 2 and 3 are equal as they to be square, the lengths of all of the metal supports are alternate interior angles. should be equal. Since I knew the length of the Since the diagonals of a rectangle are congruent and bookshelves and the distance between them, I used bisect each other, the triangle with the angles 3, 7 the Pythagorean Theorem to find the lengths of the and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. metal supports. The sum of the three angles of a triangle is 180. PROOF Write a two-column proof. Therefore, m 8 = 180 – (40 + 40) = 100. ∠ 33. Theorem 6.13 ANSWER: SOLUTION: 100 You need to walk through the proof step by step. Look over what you are given and what you need to 32. CCSS MODELING Jody is building a new prove. Here, you are given WXYZ is a rectangle with bookshelf using wood and metal supports like the one shown. To what length should she cut the metal diagonals . You need to prove supports in order for the bookshelf to be square, . Use the properties that you have learned which means that the angles formed by the shelves about rectangles to walk through the proof. and the vertical supports are all right angles? Explain your reasoning. Given: WXYZ is a rectangle with diagonals . Prove:

Proof: SOLUTION: 1. WXYZ is a rectangle with diagonals . In order for the bookshelf to be square, the lengths of (Given) all of the metal supports should be equal. Since we 2. (Opp. sides of a .) know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find 3. (Refl. Prop.) the lengths of the metal supports. 4. are right angles. (Def. of Let x be the length of each support. Express 3 feet rectangle)

as 36 inches. 5. (All right .)

6. (SAS) 7. (CPCTC)

ANSWER: Therefore, the metal supports should each be 39 Given: WXYZ is a rectangle with diagonals inches long or 3 feet 3 inches long. .

ANSWER: Prove: 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the metal supports. Proof: 1. WXYZ is a rectangle with diagonals . PROOF Write a two-column proof. (Given) 33. Theorem 6.13 2. (Opp. sides of a .) SOLUTION: 3. (Refl. Prop.) You need to walk through the proof step by step. 4. are right angles. (Def. of Look over what you are given and what you need to

prove. Here, you are given WXYZ is a rectangle with rectangle) 5. (All right .) diagonals . You need to prove 6. (SAS) . Use the properties that you have learned about rectangles to walk through the proof. 7. (CPCTC)

Given: WXYZ is a rectangle with diagonals 34. Theorem 6.14

. SOLUTION: Prove: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given . You need to prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk Proof: through the proof. 1. WXYZ is a rectangle with diagonals . (Given) Given: 2. (Opp. sides of a .) Prove: WXYZ is a rectangle. 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) 5. (All right .) 6. (SAS) Proof:

7. (CPCTC) 1. (Given) ANSWER: 2. (SSS) Given: WXYZ is a rectangle with diagonals 3. (CPCTC) . 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. Prove: sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If Proof: 2 and suppl., each is a rt. .) 1. WXYZ is a rectangle with diagonals . 9. are right angles. (If a has 1

(Given) rt. , it has 4 rt. .) 2. (Opp. sides of a .) 10. WXYZ is a rectangle. (Def. of rectangle) 3. (Refl. Prop.) ANSWER: 4. are right angles. (Def. of rectangle) Given:

5. (All right .) Prove: WXYZ is a rectangle. 6. (SAS) 7. (CPCTC)

34. Theorem 6.14 Proof: SOLUTION: 1. (Given) You need to walk through the proof step by step. Look over what you are given and what you need to 2. (SSS) prove. Here, you are given 3. (CPCTC) . You need to 4. (Def. of ) prove that WXYZ is a rectangle.. Use the properties 5. WXYZ is a parallelogram. (If both pairs of opp. that you have learned about rectangles to walk sides are , then quad. is .) through the proof. 6. are supplementary. (Cons.

are suppl.)

Given: 7. (Def of suppl.) Prove: WXYZ is a rectangle. 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

Proof: PROOF Write a paragraph proof of each 1. (Given) statement. 2. (SSS) 35. If a parallelogram has one right angle, then it is a 3. (CPCTC) rectangle. 4. (Def. of ) SOLUTION: 5. WXYZ is a parallelogram. (If both pairs of opp. You need to walk through the proof step by step. sides are , then quad. is .) Look over what you are given and what you need to 6. are supplementary. (Cons. prove. Here, you are given a parallelogram with one are suppl.) right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. 7. (Def of suppl.) Then use the properties of parallelograms to walk 8. are right angles. (If through the proof. 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle)

ANSWER:

Given: ABCD is a parallelogram, and is a right angle. Prove: WXYZ is a rectangle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle.

ANSWER:

Proof: 1. (Given) 2. (SSS) 3. (CPCTC) 4. (Def. of ) ABCD is a parallelogram, and is a right angle. 5. WXYZ is a parallelogram. (If both pairs of opp. Since ABCD is a parallelogram and has one right sides are , then quad. is .) angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. 6. are supplementary. (Cons. are suppl.) 36. If a quadrilateral has four right angles, then it is a 7. (Def of suppl.) rectangle. 8. are right angles. (If 2 and suppl., each is a rt. .) SOLUTION: You need to walk through the proof step by step. 9. are right angles. (If a has 1 Look over what you are given and what you need to rt. , it has 4 rt. .) prove. Here, you are given a quadrilateral with four 10. WXYZ is a rectangle. (Def. of rectangle) right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. PROOF Write a paragraph proof of each Then use the properties of parallelograms and statement. rectangles to walk through the proof. 35. If a parallelogram has one right angle, then it is a

rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. ABCD is a quadrilateral with four right angles. Then use the properties of parallelograms to walk ABCD is a parallelogram because both pairs of through the proof. opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. ANSWER:

ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the ABCD is a quadrilateral with four right angles. definition of a rectangle, ABCD is a rectangle. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a ANSWER: rectangle, ABCD is a rectangle.

ABCD is a parallelogram, and is a right angle. SOLUTION: Since ABCD is a parallelogram and has one right Sample answer: angle, then it has four right angles. So by the Step 1: Graph a line and place points P and Q on the definition of a rectangle, ABCD is a rectangle. line.

36. If a quadrilateral has four right angles, then it is a Step 2: Place the compass on point P. Using the rectangle. same compass setting, draw an arc to the left and SOLUTION: right of P that intersects . You need to walk through the proof step by step. Look over what you are given and what you need to Step 3: Open the compass to a setting greater than prove. Here, you are given a quadrilateral with four the distance from P to either point of intersection. right angles. You need to prove that it is a rectangle. Place the compass on each point of intersection and Let ABCD be a quadrilateral with four right angles. draw arcs that intersect above . Then use the properties of parallelograms and

rectangles to walk through the proof. Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3.

Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5. ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of Step 7: Using any compass setting, put the compass opposite angles are congruent. By definition of a at P and draw an arc above it that intersects line a. rectangle, ABCD is a rectangle. Label the point of intersection R.

ANSWER: Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw .

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. SOLUTION: Since and , the measure of Sample answer: angles P and Q is 90 degrees. Lines that are Step 1: Graph a line and place points P and Q on the perpendicular to the same line are parallel, so line. . The same compass setting was used to

locate points R and S, so . If one pair of Step 2: Place the compass on point P. Using the opposite sides of a quadrilateral is both parallel and same compass setting, draw an arc to the left and congruent, then the quadrilateral is a parallelogram. right of P that intersects . A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Place the compass on each point of intersection and ANSWER:

draw arcs that intersect above . Sample answer: Since and , the Step 4: Draw line a through point P and the measure of angles P and Q is 90 degrees. Lines that intersection of the arcs drawn in step 3. are perpendicular to the same line are parallel, so . The same compass setting was used to Step 5: Repeat steps 2 and 3 using point Q. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and Step 6: Draw line b through point Q and the congruent, then the quadrilateral is a parallelogram.

intersection of the arcs drawn in step 5. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw .

38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. SOLUTION: Sample answer: If the diagonals of a parallelogram Since and , the measure of are congruent, then it is a rectangle. He should angles P and Q is 90 degrees. Lines that are measure the diagonals of the end zone and both perpendicular to the same line are parallel, so pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a . The same compass setting was used to parallelogram. If the diagonals are congruent, then locate points R and S, so . If one pair of the end zone is a rectangle. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. ANSWER: A parallelogram with right angles is a rectangle. Sample answer: He should measure the diagonals of Thus, PRSQ is a rectangle. the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. ANSWER: ALGEBRA Quadrilateral WXYZ is a rectangle.

Sample answer: Since and , the measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so

. The same compass setting was used to 39. If XW = 3, WZ = 4, and XZ = b, find YW. locate points R and S, so . If one pair of SOLUTION: opposite sides of a quadrilateral is both parallel and The diagonals of a rectangle are congruent to each congruent, then the quadrilateral is a parallelogram. other. So, YW = XZ = b. All four angles of a A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 WZ .

Therefore, YW = 5.

ANSWER: 5

40. If XZ = 2c and ZY = 6, and XY = 8, find WY. 38. SPORTS The end zone of a football field is 160 feet SOLUTION: wide and 30 feet long. Kyle is responsible for The diagonals of a rectangle are congruent to each painting the field. He has finished the end zone. other. So, WY = XZ = 2c. All four angles of a Explain how Kyle can confirm that the end zone is rectangle are right angles. So, is a right the regulation size and be sure that it is also a 2 2 triangle. By the Pythagorean Theorem, XZ = XY + rectangle using only a tape measure. 2 ZY . SOLUTION: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then Therefore, WY = 2(5) = 10. the end zone is a rectangle. ANSWER: ANSWER: 10 Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the 41. SIGNS The sign below is in the foyer of Nyoko’s diagonals and both pairs of opposite sides are school. Based on the dimensions given, can Nyoko congruent, then the end zone is a rectangle. be sure that the sign is a rectangle? Explain your reasoning. ALGEBRA Quadrilateral WXYZ is a rectangle.

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: The diagonals of a rectangle are congruent to each SOLUTION: other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right Both pairs of opposite sides are congruent, so the triangle. By the Pythagorean Theorem, XZ2 = XW 2 + sign is a parallelogram, but no measure is given that 2 can be used to prove that it is a rectangle. If the WZ . diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle.

Therefore, YW = 5. ANSWER: ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no 5 measure is given that can be used to prove that it is a rectangle. 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. PROOF Write a coordinate proof of each SOLUTION: statement. The diagonals of a rectangle are congruent to each 42. The diagonals of a rectangle are congruent. other. So, WY = XZ = 2c. All four angles of a rectangle are right angles. So, is a right SOLUTION: Begin by positioning quadrilateral ABCD on a triangle. By the Pythagorean Theorem, XZ2 = XY2 + 2 coordinate plane. Place vertex D at the origin. Let ZY . the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to Therefore, WY = 2(5) = 10. walk through the proof. ANSWER:

10 Given: ABCD is a rectangle. Prove: 41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your reasoning.

Proof: Use the Distance Formula to find the lengths of the diagonals.

SOLUTION:

Both pairs of opposite sides are congruent, so the The diagonals, have the same length, so sign is a parallelogram, but no measure is given that they are congruent. can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the ANSWER: angles could be proven to be right angles, then we Given: ABCD is a rectangle. could prove that the sign was a rectangle. Prove: ANSWER: No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent. Proof: Use the Distance Formula to find SOLUTION: and . Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let have the same length, so they are the length of the bases be a units and the height be b congruent. units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof 43. If the diagonals of a parallelogram are congruent, step by step. Look over what you are given and what then it is a rectangle. you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the SOLUTION: properties that you have learned about rectangles to Begin by positioning parallelogram ABCD on a walk through the proof. coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c Given: ABCD is a rectangle. units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the Prove: proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Proof: Prove: ABCD is a rectangle. Use the Distance Formula to find the lengths of the diagonals.

The diagonals, have the same length, so they are congruent. Proof:

ANSWER: Given: ABCD is a rectangle. Prove: AC = BD. So,

Proof: Use the Distance Formula to find Because A and B are different points, and . a ≠ 0. Then b = 0. The slope of is undefined have the same length, so they are and the slope of . Thus, . ∠DAB is congruent. a right angle and ABCD is a rectangle.

ANSWER: 43. If the diagonals of a parallelogram are congruent, then it is a rectangle. Given: ABCD and Prove: ABCD is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given Proof: ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof. But AC = BD. So,

Given: ABCD and

Prove: ABCD is a rectangle.

Because A and B are different points,

Proof: a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the AC = BD. So, problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below. Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is c. VERBAL Make a conjecture about the diagonals a right angle and ABCD is a rectangle. of a parallelogram with four congruent sides.

ANSWER: SOLUTION: a. Given: ABCD and Sample answer: Prove: ABCD is a rectangle.

Proof:

But AC = BD. So, b. Use a protractor to measure each angle listed in the table.

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular. Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined ANSWER: and the slope of . Thus, . ∠DAB is a. Sample answer: a right angle and ABCD is a rectangle..

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. c. Sample answer: The diagonals of a parallelogram SOLUTION: with four congruent sides are perpendicular. a. Sample answer: 45. CHALLENGE In rectangle ABCD, Find the values of x and y.

b. Use a protractor to measure each angle listed in SOLUTION: the table. All four angles of a rectangle are right angles. So,

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each The diagonals of a rectangle are congruent and formed by diagonals. The diagonals of a bisect each other. So, is an isosceles triangle parallelogram with four congruent sides are with perpendicular. So,

ANSWER: a. Sample answer:

The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

ANSWER: x = 6, y = −10 b. 46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a c. Sample answer: The diagonals of a parallelogram rectangle. Tamika says that only two congruent right with four congruent sides are perpendicular. triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. 45. CHALLENGE In rectangle ABCD, SOLUTION: Find the When two congruent triangles are arranged to form a values of x and y. quadrilateral, two of the angles are formed by a single vertex of a triangle.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle. SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and

bisect each other. So, is an isosceles triangle So, Tamika is correct. with So, ANSWER: Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right The sum of the angles of a triangle is 180. angle. So, By the Vertical Angle Theorem, 47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting lines?

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right SOLUTION: triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the SOLUTION: lines intersect the line m at the points W, X, Y, and Z. When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

ANSWER: 6

OPEN ENDED So, Tamika is correct. 48. Write the equations of four lines having intersections that form the vertices of a ANSWER: rectangle. Verify your answer using coordinate Tamika; Sample answer: When two congruent geometry. triangles are arranged to form a quadrilateral, two of SOLUTION: the angles are formed by a single vertex of a triangle. Sample answer: In order for the quadrilateral to be a rectangle, one of A rectangle has 4 right angles and each pair of the angles in the congruent triangles has to be a right opposite sides is parallel and congruent. Lines that

angle. run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. 47. REASONING In the diagram at the right, lines n, p, Selecting y = 0 and x = 0 will form 2 sides of the q,and r are parallel and lines l and m are parallel. rectangle. The lines y = 4 and x = 6 form the other 2 How many rectangles are formed by the intersecting sides. lines?

x = 0, x = 6, y = 0, y = 4;

SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the The length of is 6 − 0 or 6 units and the length of lines intersect the line m at the points W, X, Y, and Z. is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

ANSWER: Sample answer: x = 0, x = 6, y = 0, Then the rectangles formed are ABXW, ACYW, y = 4; BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines.

ANSWER: 6

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: The length of is 6 − 0 or 6 units and the length of Sample answer: is 6 – 0 or 6 units. The slope of AB is 0 and the A rectangle has 4 right angles and each pair of slope of DC is 0. Since a pair of sides of the opposite sides is parallel and congruent. Lines that quadrilateral is both parallel and congruent, by run parallel to the x-axis will be parallel to each other Theorem 6.12, the quadrilateral is a parallelogram. and perpendicular to lines parallel to the y-axis. Since is horizontal and is vertical, the lines Selecting y = 0 and x = 0 will form 2 sides of the are perpendicular and the measure of the angle they rectangle. The lines y = 4 and x = 6 form the other 2 form is 90. By Theorem 6.6, if a parallelogram has 6-4 Rectanglessides. one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. x = 0, x = 6, y = 0, y = 4; 49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not rectangles. SOLUTION: Sample answer: A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always parallelograms. The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the Parallelograms with right angles are rectangles, so slope of DC is 0. Since a pair of sides of the some parallelograms are rectangles, but others with quadrilateral is both parallel and congruent, by non-right angles are not. Theorem 6.12, the quadrilateral is a parallelogram. ANSWER: Since is horizontal and is vertical, the lines Sample answer: All rectangles are parallelograms are perpendicular and the measure of the angle they because, by definition, both pairs of opposite sides form is 90. By Theorem 6.6, if a parallelogram has are parallel. Parallelograms with right angles are one right angle, it has four right angles. Therefore, by rectangles, so some parallelograms are rectangles, definition, the parallelogram is a rectangle. but others with non-right angles are not.

ANSWER: 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = Sample answer: x = 0, x = 6, y = 0, 13, what values of x and y make parallelogram y = 4; FGHJ a rectangle?

A x = 3, y = 4 B x = 4, y = 3 C x = 7, y = 8 D x = 8, y = 7 The length of is 6 − 0 or 6 units and the length of SOLUTION: is 6 – 0 or 6 units. The slope of AB is 0 and the The opposite sides of a parallelogram are congruent. slope of DC is 0. Since a pair of sides of the So, FJ = GH. quadrilateral is both parallel and congruent, by –3x + 5y = 11 FJ = GH Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines If the diagonals of a parallelogram are congruent, are perpendicular and the measure of the angle they then it is a rectangle. Since FGHJ is a parallelogram, form is 90. By Theorem 6.6, if a parallelogram has the diagonals bisect each other. So, if it is a one right angle, it has four right angles. Therefore, by rectangle, then will be an isosceles triangle definition, the parallelogram is a rectangle. with That is, 3x + y = 13.

49. WRITING IN MATH Explain why all rectangles Add the two equations to eliminate the x-term and are parallelograms, but all parallelograms are not solve for y. rectangles. –3x + 5y = 11

SOLUTION: 3x + y = 13 6y = 24 Add to eliminate x-term. Sample answer: A parallelogram is a quadrilateral in y = 4 Divide each side by 6. which each pair of opposite sides are parallel. By eSolutionsdefinitionManual of- Powered rectangle,by Cognero all rectangles have both pairs Page 19 Use the value of y to find the value of x. of opposite sides parallel. So rectangles are always 3x + 4 = 13 Substitute. parallelograms. 3x = 9 Subtract 4 from each side.

Parallelograms with right angles are rectangles, so x = 3 Divide each side by 3. some parallelograms are rectangles, but others with non-right angles are not. So, x = 3, y = 4, the correct choice is A.

ANSWER: ANSWER: Sample answer: All rectangles are parallelograms A because, by definition, both pairs of opposite sides ALGEBRA are parallel. Parallelograms with right angles are 51. A rectangular playground is surrounded rectangles, so some parallelograms are rectangles, by an 80-foot fence. One side of the playground is 10 but others with non-right angles are not. feet longer than the other. Which of the following equations could be used to find r, the shorter side of 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = the playground? 13, what values of x and y make parallelogram F 10r + r = 80 FGHJ a rectangle? G 4r + 10 = 80 H r(r + 10) = 80 J 2(r + 10) + 2r = 80 SOLUTION: The formula to find the perimeter of a rectangle is A x = 3, y = 4 given by P = 2(l + w). The perimeter is given to be 8 B x = 4, y = 3 ft. The width is r and the length is r + 10. So, 2r + 2 C x = 7, y = 8 (r + 10) = 80.

D x = 8, y = 7 Therefore, the correct choice is J. SOLUTION: ANSWER: The opposite sides of a parallelogram are congruent. J So, FJ = GH. –3x + 5y = 11 FJ = GH 52. SHORT RESPONSE What is the measure of ? If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a rectangle, then will be an isosceles triangle with That is, 3x + y = 13.

Add the two equations to eliminate the x-term and SOLUTION:

solve for y. The diagonals of a rectangle are congruent and –3x + 5y = 11 bisect each other. So, is an isosceles triangle 3x + y = 13 6y = 24 Add to eliminate x-term. with y = 4 Divide each side by 6. So, By the Exterior Angle Theorem, Use the value of y to find the value of x. 3x + 4 = 13 Substitute. ANSWER: 3x = 9 Subtract 4 from each side. x = 3 Divide each side by 3. 112

53. SAT/ACT If p is odd, which of the following must So, x = 3, y = 4, the correct choice is A. also be odd? ANSWER: A 2p A B 2p + 2

C 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 D feet longer than the other. Which of the following E p + 2 equations could be used to find r, the shorter side of the playground? SOLUTION: F 10r + r = 80 Analyze each answer choice to determine which G 4r + 10 = 80 expression is odd when p is odd. H r(r + 10) = 80 Choice A 2p J 2(r + 10) + 2r = 80 This is a multiple of 2 so it is even. Choice B 2p + 2 SOLUTION: This can be rewritten as 2(p + 1) which is also a The formula to find the perimeter of a rectangle is multiple of 2 so it is even. given by P = 2(l + w). The perimeter is given to be 8 ft. The width is r and the length is r + 10. So, 2r + 2 Choice C (r + 10) = 80. Therefore, the correct choice is J. If p is odd , is not a whole number. Choice D 2p – 2 ANSWER: This can be rewritten as 2(p - 1) which is also a J multiple of 2 so it is even. Choice E p + 2 52. SHORT RESPONSE What is the measure of Adding 2 to an odd number results in an odd ? number. This is odd. The correct choice is E.

ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So, 54. By the Exterior Angle Theorem, SOLUTION: The opposite sides of a parallelogram are congruent. ANSWER: 112

53. SAT/ACT If p is odd, which of the following must In a parallelogram, consecutive interior angles are also be odd? supplementary. A 2p B 2p + 2

C D E p + 2 SOLUTION: ANSWER: Analyze each answer choice to determine which x = 2, y = 41 expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even. 55. Choice C SOLUTION: If p is odd , is not a whole number. The opposite sides of a parallelogram are congruent. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 The opposite sides of a parallelogram are parallel to Adding 2 to an odd number results in an odd each other. So, the angles with the measures (3y + number. This is odd. 3)° and (4y – 19)° are alternate interior angles and The correct choice is E. hence they are equal.

ANSWER: E

ALGEBRA Find x and y so that the quadrilateral is a parallelogram. ANSWER: x = 8, y = 22

54. SOLUTION: 56. The opposite sides of a parallelogram are congruent. SOLUTION:

The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. In a parallelogram, consecutive interior angles are 2y = 14 supplementary. y = 7 Use the value of y in the second equation and solve for x. 7 + 3 = 2x + 6 4 = 2x 2 = x

ANSWER: ANSWER:

x = 2, y = 41 x = 2, y = 7 57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: 55. The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. SOLUTION: Find the midpoint of or . The opposite sides of a parallelogram are congruent.

Therefore, the coordinates of the intersection of the The opposite sides of a parallelogram are parallel to diagonals are (2.5, 0.5). each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and ANSWER: hence they are equal. (2.5, 0.5)

Refer to the figure.

ANSWER: x = 8, y = 22

58. If , name two congruent angles. 56. SOLUTION: SOLUTION: Since is an isosceles triangle. So, The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. ANSWER: 2y = 14 y = 7 Use the value of y in the second equation and solve 59. If , name two congruent segments. for x. 7 + 3 = 2x + 6 SOLUTION: 4 = 2x In a triangle with two congruent angles, the sides 2 = x opposite to the congruent angles are also congruent. ANSWER: So, . x = 2, y = 7 ANSWER: 57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram ABCD with vertices A(1, 3), B(6, 2), C 60. If , name two congruent segments. (4, –2), and D(–1, –1). SOLUTION: SOLUTION: In a triangle with two congruent angles, the sides The diagonals of a parallelogram bisect each other, opposite to the congruent angles are also congruent. so the intersection is the midpoint of either diagonal. So, . Find the midpoint of or . ANSWER:

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

ANSWER: FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. 3. If feet, PS = 7 feet, and SOLUTION: , find each measure. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Therefore,

ANSWER: 33.5

1. QR 4. SOLUTION: SOLUTION: The opposite sides of a rectangle are parallel and The diagonals of a rectangle are congruent and congruent. Therefore, QR = PS = 7 ft. bisect each other. So, is an isosceles triangle. Then, ANSWER: By the Vertical Angle Theorem, 7 ft Therefore, 2. SQ

SOLUTION: ANSWER: The diagonals of a rectangle bisect each other. So, 56.5

ALGEBRA Quadrilateral DEFG is a rectangle.

5. If FD = 3x – 7 and EG = x + 5, find EG. ANSWER: SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

SOLUTION: Use the value of x to find EG. The diagonals of a rectangle are congruent and EG = 6 + 5 = 11 bisect each other. So, is an isosceles triangle. Then, ANSWER: By the Exterior Angle Theorem, 11

6. If , Therefore, find . ANSWER: SOLUTION: 33.5 All four angles of a rectangle are right angles. So,

4. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem, ANSWER: 51 Therefore, 7. PROOF If ABDE is a rectangle and , ANSWER: prove that . 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: SOLUTION: You need to walk through the proof step by step. The diagonals of a rectangle are congruent to each Look over what you are given and what you need to other. So, FD = EG. prove. You are given ABDE is a rectangle and . You need to prove . Use the properties that you have learned about rectangles to walk through the proof. Use the value of x to find EG. EG = 6 + 5 = 11 Given: ABDE is a rectangle; Prove: ANSWER: 11

6. If , find .

SOLUTION: All four angles of a rectangle are right angles. So, Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle)

3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) ANSWER: 6. (SAS) 51 7. (CPCTC)

7. PROOF If ABDE is a rectangle and , ANSWER:

prove that . Given: ABDE is a rectangle; Prove:

SOLUTION: Statements(Reasons) You need to walk through the proof step by step. Look over what you are given and what you need to 1. ABDE is a rectangle; .

prove. You are given ABDE is a rectangle and (Given) 2. ABDE is a parallelogram. (Def. of . You need to prove . Use the rectangle) properties that you have learned about rectangles to walk through the proof. 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) Given: ABDE is a rectangle; 5. (All rt .)

Prove: 6. (SAS) 7. (CPCTC)

Statements(Reasons) SOLUTION: 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Use the slope formula to find the slope of the sides of Given: ABDE is a rectangle; the quadrilateral. Prove:

Statements(Reasons) 1. ABDE is a rectangle; . There are no parallel sides, so WXYZ is not a (Given) parallelogram. Therefore, WXYZ is not a rectangle. 2. ABDE is a parallelogram. (Def. of rectangle) ANSWER: 3. (Opp. sides of a .) No; slope of , slope of , slope of 4. are right angles.(Def. of rectangle) 5. (All rt .) , and slope of . Slope of 6. (SAS) 7. (CPCTC) slope of , and slope of slope of , so WXYZ is not a parallelogram. Therefore, COORDINATE GEOMETRY Graph each WXYZ is not a rectangle. quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

Use the slope formula to find the slope of the sides of the quadrilateral.

First, Use the Distance formula to find the lengths of There are no parallel sides, so WXYZ is not a the sides of the quadrilateral. parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of so ABCD is a , and slope of . Slope of parallelogram. A parallelogram is a rectangle if the diagonals are slope of , and slope of slope of congruent. Use the Distance formula to find the , so WXYZ is not a parallelogram. Therefore, lengths of the diagonals. WXYZ is not a rectangle.

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so ABCD is a 10. BC

parallelogram. SOLUTION: A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the The opposite sides of a rectangle are parallel and

lengths of the diagonals. congruent. Therefore, BC = AD = 2 ft. ANSWER: 2 ft So the diagonals are congruent. Thus, ABCD is a rectangle. 11. DB ANSWER: SOLUTION: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a All four angles of a rectangle are right angles. So, parallelogram. , so the diagonals are is a right triangle. By the Pythagorean 2 2 2 congruent. Thus, ABCD is a rectangle. Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

ANSWER: 6.3 ft

12. SOLUTION: FENCING X-braces are also used to provide The diagonals of a rectangle are congruent and support in rectangular fencing. If AB = 6 feet, bisect each other. So, is an isosceles AD = 2 feet, and , find each triangle. measure. Then,

By the Vertical Angle Theorem,

ANSWER: 50

10. BC 13. SOLUTION: SOLUTION: The opposite sides of a rectangle are parallel and The diagonals of a rectangle are congruent and congruent. Therefore, BC = AD = 2 ft. bisect each other. So, is an isosceles triangle. ANSWER: Then, 2 ft All four angles of a rectangle are right angles.

11. DB SOLUTION: All four angles of a rectangle are right angles. So, ANSWER: is a right triangle. By the Pythagorean 25 2 2 2 Theorem, DB = BC + DC . CCSS REGULARITY Quadrilateral WXYZ is a

BC = AD = 2 rectangle. DC = AB = 6

ANSWER: 6.3 ft 14. If ZY = 2x + 3 and WX = x + 4, find WX.

12. SOLUTION: The opposite sides of a rectangle are parallel and SOLUTION: congruent. So, ZY = WX. The diagonals of a rectangle are congruent and 2x + 3 = x + 4 bisect each other. So, is an isosceles x = 1 triangle. Use the value of x to find WX. Then, WX = 1 + 4 = 5.

ANSWER: 5 By the Vertical Angle Theorem, 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. ANSWER: SOLUTION: 50 The diagonals of a rectangle bisect each other. So, PY = WP. 13. SOLUTION: The diagonals of a rectangle are congruent and The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. bisect each other. So, is an isosceles triangle. Then, All four angles of a rectangle are right angles.

ANSWER: 43

ANSWER: 16. If , find 25 . SOLUTION: CCSS REGULARITY Quadrilateral WXYZ is a rectangle. All four angles of a rectangle are right angles. So,

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: ANSWER: The opposite sides of a rectangle are parallel and 39 congruent. So, ZY = WX. 2x + 3 = x + 4 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX.

x = 1 SOLUTION: Use the value of x to find WX. The diagonals of a rectangle are congruent and WX = 1 + 4 = 5. bisect each other. So, ZP = PY. ANSWER: 5

15. If PY = 3x – 5 and WP = 2x + 11, find ZP. Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38. SOLUTION: The diagonals of a rectangle bisect each other. So, ANSWER: PY = WP. 38

18. If , find The diagonals of a rectangle are congruent and . bisect each other. So, is an isosceles triangle. SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 43

16. If , find . ANSWER: 48 SOLUTION: All four angles of a rectangle are right angles. So, 19. If , find .

SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. Then,

ANSWER: 39 Therefore, Since is a 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. transversal of parallel sides and , alternate SOLUTION: interior angles WZX and ZXY are congruent. So, The diagonals of a rectangle are congruent and = = 46. bisect each other. So, ZP = PY. ANSWER: 46 PROOF Write a two-column proof. Then ZP = 4(7) – 9 = 19. 20. Given: ABCD is a rectangle. Therefore, ZX = 2(ZP) = 38. Prove: ANSWER: 38

18. If , find

. SOLUTION: SOLUTION: You need to walk through the proof step by step. All four angles of a rectangle are right angles. So, Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove: ANSWER: Proof: 48 Statements (Reasons) 1. ABCD is a rectangle. (Given) 19. If , find 2. ABCD is a parallelogram. (Def. of rectangle)

. 3. (Opp. sides of a .) SOLUTION: 4. (Refl. Prop.) All four angles of a rectangle are right angles. So, 5. (Diagonals of a rectangle are .) is a right triangle. Then, 6. (SSS)

ANSWER: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) Therefore, Since is a 2. ABCD is a parallelogram. (Def. of rectangle) transversal of parallel sides and , alternate 3. (Opp. sides of a .) interior angles WZX and ZXY are congruent. So, 4. (Refl. Prop.) = = 46. 5. (Diagonals of a rectangle are .) ANSWER: 6. (SSS) 46 21. Given: QTVW is a rectangle. PROOF Write a two-column proof.

20. Given: ABCD is a rectangle. Prove:

Prove:

SOLUTION: SOLUTION: You need to walk through the proof step by step. You need to walk through the proof step by step. Look over what you are given and what you need to Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. prove. Here, you are given QTVW is a rectangle and You need to prove . Use the . You need to prove . Use properties that you have learned about rectangles to the properties that you have learned about rectangles walk through the proof. to walk through the proof.

Given: ABCD is a rectangle. Given: QTVW is a rectangle. Prove: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .)

4. (Refl. Prop.) Prove: Proof: 5. (Diagonals of a rectangle are .) Statements (Reasons) 6. (SSS) 1. QTVW is a rectangle; .(Given) ANSWER: 2. QTVW is a parallelogram. (Def. of rectangle) Proof: 3. (Opp sides of a .) Statements (Reasons) 4. are right angles.(Def. of rectangle) 1. ABCD is a rectangle. (Given) 5. (All rt .) 2. ABCD is a parallelogram. (Def. of rectangle) 6. QR = ST (Def. of segs.) 3. (Opp. sides of a .) 7. (Refl. Prop.) 4. (Refl. Prop.)

8. RS = RS (Def. of segs.) 5. (Diagonals of a rectangle are .) 9. QR + RS = RS + ST (Add. prop.) 6. (SSS) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 21. Given: QTVW is a rectangle. 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) Prove: 2. QTVW is a parallelogram. (Def. of rectangle) SOLUTION: 3. (Opp sides of a .) You need to walk through the proof step by step. 4. are right angles.(Def. of rectangle) Look over what you are given and what you need to 5. (All rt .) prove. Here, you are given QTVW is a rectangle and 6. QR = ST (Def. of segs.) . You need to prove . Use 7. (Refl. Prop.) the properties that you have learned about rectangles

to walk through the proof. 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) Given: QTVW is a rectangle. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

11. QS = RT (Subst.) 12. (Def. of segs.) Use the slope formula to find the slope of the sides of 13. (SAS) the quadrilateral.

ANSWER: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) The slopes of each pair of opposite sides are equal. 6. QR = ST (Def. of segs.) So, the two pairs of opposite sides are parallel. 7. (Refl. Prop.) Therefore, the quadrilateral WXYZ is a parallelogram. 8. RS = RS (Def. of segs.) The products of the slopes of the adjacent sides are 9. QR + RS = RS + ST (Add. prop.) –1. So, any two adjacent sides are perpendicular to 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) each other. That is, all the four angles are right 11. QS = RT (Subst.) angles. Therefore, WXYZ is a rectangle. 12. (Def. of segs.) ANSWER: 13. (SAS) Yes; slope of slope of , slope COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine of slope of . So WXYZ is a whether the figure is a rectangle. Justify your parallelogram. The product of the slopes of answer using the indicated formula. consecutive sides is –1, so the consecutive sides are 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula perpendicular and form right angles. Thus, WXYZ is a rectangle. SOLUTION:

Use the slope formula to find the slope of the sides of 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance

the quadrilateral. Formula

SOLUTION:

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right First, Use the Distance formula to find the lengths of angles. Therefore, WXYZ is a rectangle. the sides of the quadrilateral.

ANSWER:

Yes; slope of slope of , slope

of slope of . So WXYZ is a so JKLM is a parallelogram. The product of the slopes of parallelogram. consecutive sides is –1, so the consecutive sides are A parallelogram is a rectangle if the diagonals are perpendicular and form right angles. Thus, WXYZ is a congruent. Use the Distance formula to find the rectangle. lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the First, use the Distance Formula to find the lengths of lengths of the diagonals. the sides of the quadrilateral.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: so QRST is a parallelogram. No; so JKLM is a A parallelogram is a rectangle if the diagonals are parallelogram; so congruent. Use the Distance Formula to find the the diagonals are not congruent. Thus, JKLM is not a lengths of the diagonals. rectangle.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle. SOLUTION:

First, use the slope formula to find the lengths of the Quadrilateral ABCD is a rectangle. Find each sides of the quadrilateral. measure if .

26. SOLUTION: All four angles of a rectangle are right angles. So, The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a parallelogram. None of the adjacent sides have slopes whose ANSWER: product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle. 50

ANSWER: 27. No; slope of slope of and slope of SOLUTION: The measures of angles 2 and 3 are equal as they slope of . So, GHJK is a are alternate interior angles. parallelogram. The product of the slopes of Since the diagonals of a rectangle are congruent and consecutive sides ≠ –1, so the consecutive sides are bisect each other, the triangle with the angles 3, 7 not perpendicular. Thus, GHJK is not a rectangle. and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: 40

28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40.

ANSWER: 40 Quadrilateral ABCD is a rectangle. Find each measure if . 29. SOLUTION: All four angles of a rectangle are right angles. So,

26. The measures of angles 1 and 4 are equal as they

SOLUTION: are alternate interior angles. Therefore, m 4 = m 1 = 50. All four angles of a rectangle are right angles. So, ∠ ∠ Since the diagonals of a rectangle are congruent and

bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. ANSWER: 50 ANSWER: 80 27. 30. SOLUTION: SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. All four angles of a rectangle are right angles. So, Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. The measures of angles 1 and 4 are equal as they are alternate interior angles. ANSWER: Therefore, m∠4 = m∠1 = 50. 40 Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 28. and 6 is an isosceles triangle. Therefore, m∠6 = SOLUTION: m∠4 = 50. The measures of angles 2 and 3 are equal as they ANSWER: are alternate interior angles. 50 Therefore, m∠3 = m∠2 = 40. ANSWER: 31. 40 SOLUTION: The measures of angles 2 and 3 are equal as they 29. are alternate interior angles. SOLUTION: Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 All four angles of a rectangle are right angles. So, and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180.

Therefore, m∠8 = 180 – (40 + 40) = 100. The measures of angles 1 and 4 are equal as they ANSWER: are alternate interior angles. 100 Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and 32. CCSS MODELING Jody is building a new bisect each other, the triangle with the angles 4, 5 bookshelf using wood and metal supports like the one and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. shown. To what length should she cut the metal The sum of the three angles of a triangle is 180. supports in order for the bookshelf to be square, Therefore, m∠5 = 180 – (50 + 50) = 80. which means that the angles formed by the shelves and the vertical supports are all right angles? Explain ANSWER: your reasoning. 80

30. SOLUTION: All four angles of a rectangle are right angles. So,

The measures of angles 1 and 4 are equal as they SOLUTION: are alternate interior angles. In order for the bookshelf to be square, the lengths of Therefore, m∠4 = m∠1 = 50. all of the metal supports should be equal. Since we Since the diagonals of a rectangle are congruent and know the length of the bookshelves and the distance bisect each other, the triangle with the angles 4, 5 between them, use the Pythagorean Theorem to find

and 6 is an isosceles triangle. Therefore, m∠6 = the lengths of the metal supports. Let x be the length of each support. Express 3 feet m∠4 = 50. as 36 inches. ANSWER: 50

31. SOLUTION: Therefore, the metal supports should each be 39 The measures of angles 2 and 3 are equal as they inches long or 3 feet 3 inches long. are alternate interior angles. Since the diagonals of a rectangle are congruent and ANSWER: bisect each other, the triangle with the angles 3, 7 3 ft 3 in.; Sample answer: In order for the bookshelf and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. to be square, the lengths of all of the metal supports The sum of the three angles of a triangle is 180. should be equal. Since I knew the length of the Therefore, m∠8 = 180 – (40 + 40) = 100. bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the ANSWER: metal supports. 100 PROOF Write a two-column proof. 32. CCSS MODELING Jody is building a new 33. Theorem 6.13 bookshelf using wood and metal supports like the one SOLUTION: shown. To what length should she cut the metal supports in order for the bookshelf to be square, You need to walk through the proof step by step. which means that the angles formed by the shelves Look over what you are given and what you need to and the vertical supports are all right angles? Explain prove. Here, you are given WXYZ is a rectangle with your reasoning. diagonals . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: WXYZ is a rectangle with diagonals . Prove:

SOLUTION: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we know the length of the bookshelves and the distance Proof: between them, use the Pythagorean Theorem to find 1. WXYZ is a rectangle with diagonals . the lengths of the metal supports.

Let x be the length of each support. Express 3 feet (Given) as 36 inches. 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) 5. (All right .)

Therefore, the metal supports should each be 39 6. (SAS) inches long or 3 feet 3 inches long. 7. (CPCTC)

ANSWER: ANSWER: 3 ft 3 in.; Sample answer: In order for the bookshelf Given: WXYZ is a rectangle with diagonals to be square, the lengths of all of the metal supports . should be equal. Since I knew the length of the bookshelves and the distance between them, I used Prove: the Pythagorean Theorem to find the lengths of the metal supports.

PROOF Write a two-column proof. 33. Theorem 6.13 SOLUTION: Proof: You need to walk through the proof step by step. 1. WXYZ is a rectangle with diagonals . Look over what you are given and what you need to (Given) prove. Here, you are given WXYZ is a rectangle with 2. (Opp. sides of a .) diagonals . You need to prove 3. (Refl. Prop.) . Use the properties that you have learned 4. are right angles. (Def. of about rectangles to walk through the proof. rectangle) 5. (All right .) Given: WXYZ is a rectangle with diagonals 6. (SAS) . 7. (CPCTC) Prove: 34. Theorem 6.14 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to Proof: prove. Here, you are given 1. WXYZ is a rectangle with diagonals . . You need to (Given) prove that WXYZ is a rectangle.. Use the properties 2. (Opp. sides of a .) that you have learned about rectangles to walk through the proof. 3. (Refl. Prop.)

4. are right angles. (Def. of Given: rectangle) Prove: WXYZ is a rectangle. 5. (All right .) 6. (SAS) 7. (CPCTC)

ANSWER:

Given: WXYZ is a rectangle with diagonals Proof:

. 1. (Given) Prove: 2. (SSS) 3. (CPCTC) 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .)

Proof: 6. are supplementary. (Cons. 1. WXYZ is a rectangle with diagonals . are suppl.) (Given) 7. (Def of suppl.) 2. (Opp. sides of a .) 8. are right angles. (If

3. (Refl. Prop.) 2 and suppl., each is a rt. .) 4. are right angles. (Def. of 9. are right angles. (If a has 1 rectangle) rt. , it has 4 rt. .)

5. (All right .) 10. WXYZ is a rectangle. (Def. of rectangle) 6. (SAS) ANSWER: 7. (CPCTC) Given: Prove: WXYZ is a rectangle. 34. Theorem 6.14 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given Proof: . You need to 1. (Given) prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk 2. (SSS) through the proof. 3. (CPCTC) 4. (Def. of ) Given: 5. WXYZ is a parallelogram. (If both pairs of opp. Prove: WXYZ is a rectangle. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If 2 and suppl., each is a rt. .) Proof: 9. are right angles. (If a has 1 1. (Given) rt. , it has 4 rt. .)

2. (SSS) 10. WXYZ is a rectangle. (Def. of rectangle) 3. (CPCTC) 4. (Def. of ) PROOF Write a paragraph proof of each 5. WXYZ is a parallelogram. (If both pairs of opp. statement. 35. If a parallelogram has one right angle, then it is a sides are , then quad. is .) rectangle. 6. are supplementary. (Cons. are suppl.) SOLUTION: 7. (Def of suppl.) You need to walk through the proof step by step. Look over what you are given and what you need to 8. are right angles. (If prove. Here, you are given a parallelogram with one 2 and suppl., each is a rt. .) right angle. You need to prove that it is a rectangle. 9. are right angles. (If a has 1 Let ABCD be a parallelogram with one right angle. rt. , it has 4 rt. .) Then use the properties of parallelograms to walk 10. WXYZ is a rectangle. (Def. of rectangle) through the proof.

ANSWER: Given: Prove: WXYZ is a rectangle.

ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the Proof: definition of a rectangle, ABCD is a rectangle. 1. (Given) ANSWER: 2. (SSS) 3. (CPCTC) 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right

7. (Def of suppl.) angle, then it has four right angles. So by the 8. are right angles. (If definition of a rectangle, ABCD is a rectangle. 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 36. If a quadrilateral has four right angles, then it is a rt. , it has 4 rt. .) rectangle. 10. WXYZ is a rectangle. (Def. of rectangle) SOLUTION: PROOF Write a paragraph proof of each You need to walk through the proof step by step. statement. Look over what you are given and what you need to 35. If a parallelogram has one right angle, then it is a prove. Here, you are given a quadrilateral with four rectangle. right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. SOLUTION: Then use the properties of parallelograms and You need to walk through the proof step by step. rectangles to walk through the proof. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

ANSWER: ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle.

ANSWER: ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle.

37. CONSTRUCTION Construct a rectangle using the ABCD is a parallelogram, and is a right angle. construction for congruent segments and the Since ABCD is a parallelogram and has one right construction for a line perpendicular to another line angle, then it has four right angles. So by the through a point on the line. Justify each step of the definition of a rectangle, ABCD is a rectangle. construction. SOLUTION: 36. If a quadrilateral has four right angles, then it is a Sample answer: rectangle. Step 1: Graph a line and place points P and Q on the SOLUTION: line. You need to walk through the proof step by step. Look over what you are given and what you need to Step 2: Place the compass on point P. Using the prove. Here, you are given a quadrilateral with four same compass setting, draw an arc to the left and right angles. You need to prove that it is a rectangle. right of P that intersects . Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and Step 3: Open the compass to a setting greater than rectangles to walk through the proof. the distance from P to either point of intersection. Place the compass on each point of intersection and draw arcs that intersect above .

Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3.

ANSWER: Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a Step 9: Draw . rectangle, ABCD is a rectangle.

37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line. Justify each step of the construction. SOLUTION: Sample answer: Step 1: Graph a line and place points P and Q on the line. Since and , the measure of Step 2: Place the compass on point P. Using the angles P and Q is 90 degrees. Lines that are same compass setting, draw an arc to the left and perpendicular to the same line are parallel, so right of P that intersects . . The same compass setting was used to locate points R and S, so . If one pair of Step 3: Open the compass to a setting greater than opposite sides of a quadrilateral is both parallel and the distance from P to either point of intersection. congruent, then the quadrilateral is a parallelogram. Place the compass on each point of intersection and A parallelogram with right angles is a rectangle. draw arcs that intersect above . Thus, PRSQ is a rectangle.

Step 4: Draw line a through point P and the ANSWER: intersection of the arcs drawn in step 3.

Step 7: Using any compass setting, put the compass locate points R and S, so . If one pair of at P and draw an arc above it that intersects line a. opposite sides of a quadrilateral is both parallel and Label the point of intersection R. congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle. Step 8: Using the same compass setting, put the compass at Q and draw an arc above it that intersects line b. Label the point of intersection S.

Step 9: Draw .

38. SPORTS The end zone of a football field is 160 feet wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a Since and , the measure of rectangle using only a tape measure. angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so SOLUTION: . The same compass setting was used to Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should locate points R and S, so . If one pair of measure the diagonals of the end zone and both opposite sides of a quadrilateral is both parallel and pairs of opposite sides. If both pairs of opposite sides congruent, then the quadrilateral is a parallelogram. are congruent, then the quadrilateral is a A parallelogram with right angles is a rectangle. parallelogram. If the diagonals are congruent, then Thus, PRSQ is a rectangle. the end zone is a rectangle.

ANSWER: ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are Sample answer: Since and , the congruent, then the end zone is a rectangle. measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so ALGEBRA Quadrilateral WXYZ is a rectangle. . The same compass setting was used to locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. 39. If XW = 3, WZ = 4, and XZ = b, find YW. Thus, PRSQ is a rectangle. SOLUTION: The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean Theorem, XZ2 = XW 2 + 2 WZ .

Therefore, YW = 5.

ANSWER: 38. SPORTS The end zone of a football field is 160 feet 5 wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. Explain how Kyle can confirm that the end zone is 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. the regulation size and be sure that it is also a SOLUTION:

rectangle using only a tape measure. The diagonals of a rectangle are congruent to each SOLUTION: other. So, WY = XZ = 2c. All four angles of a Sample answer: If the diagonals of a parallelogram rectangle are right angles. So, is a right are congruent, then it is a rectangle. He should triangle. By the Pythagorean Theorem, XZ2 = XY2 + 2 measure the diagonals of the end zone and both ZY . pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. If the diagonals are congruent, then the end zone is a rectangle.

ANSWER: Sample answer: He should measure the diagonals of Therefore, WY = 2(5) = 10. the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are ANSWER: congruent, then the end zone is a rectangle. 10

ALGEBRA Quadrilateral WXYZ is a rectangle. 41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your reasoning.

Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that Therefore, YW = 5. can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the ANSWER: angles could be proven to be right angles, then we 5 could prove that the sign was a rectangle. ANSWER: 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. No; sample answer: Both pairs of opposite sides are SOLUTION: congruent, so the sign is a parallelogram, but no The diagonals of a rectangle are congruent to each measure is given that can be used to prove that it is a other. So, WY = XZ = 2c. All four angles of a rectangle. rectangle are right angles. So, is a right PROOF Write a coordinate proof of each 2 2 triangle. By the Pythagorean Theorem, XZ = XY + statement. 2 ZY . 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, Therefore, WY = 2(5) = 10. b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what ANSWER: you need to prove. Here, you are given ABCD is a 10 rectangle and you need to prove . Use the properties that you have learned about rectangles to 41. SIGNS The sign below is in the foyer of Nyoko’s walk through the proof. school. Based on the dimensions given, can Nyoko be sure that the sign is a rectangle? Explain your Given: ABCD is a rectangle. reasoning. Prove:

SOLUTION: Proof: Use the Distance Formula to find the lengths of the

diagonals. Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle. The diagonals, have the same length, so they are congruent. ANSWER: No; sample answer: Both pairs of opposite sides are ANSWER: congruent, so the sign is a parallelogram, but no Given: ABCD is a rectangle. measure is given that can be used to prove that it is a Prove: rectangle.

PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a

coordinate plane. Place vertex D at the origin. Let Proof: the length of the bases be a units and the height be b Use the Distance Formula to find units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof and . step by step. Look over what you are given and what have the same length, so they are you need to prove. Here, you are given ABCD is a congruent. rectangle and you need to prove . Use the properties that you have learned about rectangles to 43. If the diagonals of a parallelogram are congruent,

walk through the proof. then it is a rectangle.

Given: ABCD is a rectangle. SOLUTION: Prove: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you Proof: have learned about rectangles to walk through the Use the Distance Formula to find the lengths of the proof. diagonals.

Given: ABCD and Prove: ABCD is a rectangle.

The diagonals, have the same length, so they are congruent.

ANSWER: Given: ABCD is a rectangle.

Prove: Proof:

AC = BD. So,

Proof: Use the Distance Formula to find and . have the same length, so they are congruent. Because A and B are different points,

43. If the diagonals of a parallelogram are congruent, a ≠ 0. Then b = 0. The slope of is undefined then it is a rectangle. and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a ANSWER: coordinate plane. Place vertex A at the origin. Let Given: ABCD and the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + Prove: ABCD is a rectangle. a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof. Proof:

Given: ABCD and Prove: ABCD is a rectangle.

But AC = BD. So,

Proof:

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined AC = BD. So, and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Because A and B are different points, Draw the two diagonals of each parallelogram and a ≠ 0. Then b = 0. The slope of is undefined label the intersections R. and the slope of . Thus, . ∠DAB is b. TABULAR Use a protractor to measure the a right angle and ABCD is a rectangle. appropriate angles and complete the table below.

ANSWER:

Given: ABCD and c. VERBAL Make a conjecture about the diagonals Prove: ABCD is a rectangle. of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

Proof:

But AC = BD. So,

b. Use a protractor to measure each angle listed in the table.

Because A and B are different points, c. Sample answer: Each of the angles listed in the table are right angles. These angles were each a ≠ 0. Then b = 0. The slope of is undefined formed by diagonals. The diagonals of a and the slope of . Thus, . ∠DAB is parallelogram with four congruent sides are a right angle and ABCD is a rectangle.. perpendicular.

44. MULTIPLE REPRESENTATIONS In the ANSWER: problem, you will explore properties of other special a. Sample answer: parallelograms. a. GEOMETRIC Draw three parallelograms, each with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R.

b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides.

SOLUTION: b. a. Sample answer:

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the

values of x and y.

b. Use a protractor to measure each angle listed in the table.

ANSWER: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle a. Sample answer: with So,

The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

ANSWER: c. Sample answer: The diagonals of a parallelogram x = 6, y = −10 with four congruent sides are perpendicular. 46. CCSS CRITIQUE Parker says that any two 45. CHALLENGE In rectangle ABCD, congruent acute triangles can be arranged to make a Find the rectangle. Tamika says that only two congruent right values of x and y. triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

SOLUTION: All four angles of a rectangle are right angles. So, In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So,

So, Tamika is correct.

ANSWER: The sum of the angles of a triangle is 180. Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of So, the angles are formed by a single vertex of a triangle. By the Vertical Angle Theorem, In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

47. REASONING In the diagram at the right, lines n, p, q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting ANSWER: lines? x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a SOLUTION: quadrilateral, two of the angles are formed by a Let the lines n, p, q, and r intersect the line l at the single vertex of a triangle. points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 rectangles are formed by the intersecting lines. So, Tamika is correct. ANSWER: ANSWER: 6 Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of 48. OPEN ENDED Write the equations of four lines the angles are formed by a single vertex of a triangle. having intersections that form the vertices of a In order for the quadrilateral to be a rectangle, one of rectangle. Verify your answer using coordinate the angles in the congruent triangles has to be a right geometry. angle. SOLUTION: 47. REASONING In the diagram at the right, lines n, p, Sample answer: q,and r are parallel and lines l and m are parallel. A rectangle has 4 right angles and each pair of How many rectangles are formed by the intersecting opposite sides is parallel and congruent. Lines that lines? run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

x = 0, x = 6, y = 0, y = 4;

SOLUTION: Let the lines n, p, q, and r intersect the line l at the points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other The length of is 6 − 0 or 6 units and the length of and perpendicular to lines parallel to the y-axis. is 6 – 0 or 6 units. The slope of AB is 0 and the Selecting y = 0 and x = 0 will form 2 sides of the slope of DC is 0. Since a pair of sides of the rectangle. The lines y = 4 and x = 6 form the other 2 quadrilateral is both parallel and congruent, by sides. Theorem 6.12, the quadrilateral is a parallelogram.

Since is horizontal and is vertical, the lines x = 0, x = 6, y = 0, y = 4; are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not rectangles. SOLUTION: Sample answer: A parallelogram is a quadrilateral in The length of is 6 − 0 or 6 units and the length of which each pair of opposite sides are parallel. By is 6 – 0 or 6 units. The slope of AB is 0 and the definition of rectangle, all rectangles have both pairs slope of DC is 0. Since a pair of sides of the of opposite sides parallel. So rectangles are always quadrilateral is both parallel and congruent, by parallelograms. Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines Parallelograms with right angles are rectangles, so are perpendicular and the measure of the angle they some parallelograms are rectangles, but others with form is 90. By Theorem 6.6, if a parallelogram has non-right angles are not. one right angle, it has four right angles. Therefore, by ANSWER: definition, the parallelogram is a rectangle. Sample answer: All rectangles are parallelograms ANSWER: because, by definition, both pairs of opposite sides Sample answer: x = 0, x = 6, y = 0, are parallel. Parallelograms with right angles are y = 4; rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram FGHJ a rectangle?

A The length of is 6 − 0 or 6 units and the length of x = 3, y = 4 B x = 4, y = 3 is 6 – 0 or 6 units. The slope of AB is 0 and the C x = 7, y = 8 slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by D x = 8, y = 7 Theorem 6.12, the quadrilateral is a parallelogram. SOLUTION: Since is horizontal and is vertical, the lines The opposite sides of a parallelogram are congruent. are perpendicular and the measure of the angle they So, FJ = GH. form is 90. By Theorem 6.6, if a parallelogram has –3x + 5y = 11 FJ = GH one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle. If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, 49. WRITING IN MATH Explain why all rectangles the diagonals bisect each other. So, if it is a are parallelograms, but all parallelograms are not rectangle, then will be an isosceles triangle

rectangles. with That is, 3x + y = 13. SOLUTION: Sample answer: A parallelogram is a quadrilateral in Add the two equations to eliminate the x-term and which each pair of opposite sides are parallel. By solve for y. definition of rectangle, all rectangles have both pairs –3x + 5y = 11 of opposite sides parallel. So rectangles are always 3x + y = 13 parallelograms. 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6. Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with Use the value of y to find the value of x. non-right angles are not. 3x + 4 = 13 Substitute. 3x = 9 Subtract 4 from each side. ANSWER: x = 3 Divide each side by 3. Sample answer: All rectangles are parallelograms because, by definition, both pairs of opposite sides So, x = 3, y = 4, the correct choice is A. are parallel. Parallelograms with right angles are 6-4 Rectanglesrectangles, so some parallelograms are rectangles, ANSWER: but others with non-right angles are not. A

50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 51. ALGEBRA A rectangular playground is surrounded 13, what values of x and y make parallelogram by an 80-foot fence. One side of the playground is 10 FGHJ a rectangle? feet longer than the other. Which of the following equations could be used to find r, the shorter side of the playground? F 10r + r = 80 G 4r + 10 = 80 H r(r + 10) = 80 A x = 3, y = 4 J 2(r + 10) + 2r = 80 B x = 4, y = 3 C x = 7, y = 8 SOLUTION: D x = 8, y = 7 The formula to find the perimeter of a rectangle is given by P = 2(l + w). The perimeter is given to be 8 SOLUTION: ft. The width is r and the length is r + 10. So, 2r + 2 The opposite sides of a parallelogram are congruent. (r + 10) = 80. So, FJ = GH. Therefore, the correct choice is J. –3x + 5y = 11 FJ = GH ANSWER: If the diagonals of a parallelogram are congruent, J then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a 52. SHORT RESPONSE What is the measure of rectangle, then will be an isosceles triangle ? with That is, 3x + y = 13.

Add the two equations to eliminate the x-term and solve for y. –3x + 5y = 11 3x + y = 13 6y = 24 Add to eliminate x-term. SOLUTION: y = 4 Divide each side by 6. The diagonals of a rectangle are congruent and Use the value of y to find the value of x. bisect each other. So, is an isosceles triangle 3x + 4 = 13 Substitute. with 3x = 9 Subtract 4 from each side. So, x = 3 Divide each side by 3. By the Exterior Angle Theorem,

So, x = 3, y = 4, the correct choice is A. ANSWER: ANSWER: 112 A 53. SAT/ACT If p is odd, which of the following must 51. ALGEBRA A rectangular playground is surrounded also be odd? by an 80-foot fence. One side of the playground is 10 A 2p feet longer than the other. Which of the following B 2p + 2 equations could be used to find r, the shorter side of the playground? C F 10r + r = 80 D G 4r + 10 = 80 E p + 2 H r(r + 10) = 80 J 2(r + 10) + 2r = 80 SOLUTION: Analyze each answer choice to determine which SOLUTION: expression is odd when p is odd. eSolutionsThe Manualformula- Powered to findby theCognero perimeter of a rectangle is Choice A 2p Page 20 given by P = 2(l + w). The perimeter is given to be 8 This is a multiple of 2 so it is even. ft. The width is r and the length is r + 10. So, 2r + 2 Choice B 2p + 2 (r + 10) = 80. This can be rewritten as 2(p + 1) which is also a Therefore, the correct choice is J. multiple of 2 so it is even. ANSWER: Choice C J If p is odd , is not a whole number. 52. SHORT RESPONSE What is the measure of Choice D 2p – 2 ? This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. The correct choice is E.

SOLUTION: ANSWER: The diagonals of a rectangle are congruent and E bisect each other. So, is an isosceles triangle ALGEBRA Find x and y so that the with quadrilateral is a parallelogram. So, By the Exterior Angle Theorem,

ANSWER: 112 54. SOLUTION: 53. SAT/ACT If p is odd, which of the following must The opposite sides of a parallelogram are congruent. also be odd?

A 2p B 2p + 2

C In a parallelogram, consecutive interior angles are supplementary. D E p + 2 SOLUTION: Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. ANSWER: Choice B 2p + 2 x = 2, y = 41 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

Choice C If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a 55. multiple of 2 so it is even. SOLUTION: Choice E p + 2 The opposite sides of a parallelogram are congruent. Adding 2 to an odd number results in an odd number. This is odd. The correct choice is E.

ANSWER: The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + E 3)° and (4y – 19)° are alternate interior angles and hence they are equal. ALGEBRA Find x and y so that the

quadrilateral is a parallelogram.

ANSWER: 54. x = 8, y = 22 SOLUTION: The opposite sides of a parallelogram are congruent.

56. In a parallelogram, consecutive interior angles are SOLUTION: supplementary. The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 Use the value of y in the second equation and solve for x. ANSWER: 7 + 3 = 2x + 6 x = 2, y = 41 4 = 2x 2 = x

ANSWER: x = 2, y = 7

57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of 55. parallelogram ABCD with vertices A(1, 3), B(6, 2), C SOLUTION: (4, –2), and D(–1, –1). The opposite sides of a parallelogram are congruent. SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or . The opposite sides of a parallelogram are parallel to each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and hence they are equal. Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: (2.5, 0.5)

ANSWER: Refer to the figure. x = 8, y = 22

56. SOLUTION: The diagonals of a parallelogram bisect each other. 58. If , name two congruent angles. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. SOLUTION: 2y = 14 Since is an isosceles triangle. So, y = 7 Use the value of y in the second equation and solve for x. ANSWER: 7 + 3 = 2x + 6 4 = 2x 2 = x 59. If , name two congruent segments. ANSWER: SOLUTION: x = 2, y = 7 In a triangle with two congruent angles, the sides 57. COORDINATE GEOMETRY Find the opposite to the congruent angles are also congruent. coordinates of the intersection of the diagonals of So, . parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). ANSWER:

SOLUTION: The diagonals of a parallelogram bisect each other, 60. If , name two congruent segments. so the intersection is the midpoint of either diagonal. Find the midpoint of or . SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, . Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5). ANSWER: ANSWER: (2.5, 0.5)

Refer to the figure.

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

ANSWER:

3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, 1. QR By the Exterior Angle Theorem, SOLUTION:

The opposite sides of a rectangle are parallel and Therefore, congruent. Therefore, QR = PS = 7 ft.

ANSWER: ANSWER: 7 ft 33.5

4. 2. SQ SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and The diagonals of a rectangle bisect each other. So, bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem,

Therefore,

ANSWER: 56.5 ANSWER: ALGEBRA Quadrilateral DEFG is a rectangle.

3. SOLUTION: The diagonals of a rectangle are congruent and 5. If FD = 3x – 7 and EG = x + 5, find EG. bisect each other. So, is an isosceles triangle. SOLUTION: Then, The diagonals of a rectangle are congruent to each By the Exterior Angle Theorem, other. So, FD = EG.

Therefore, Use the value of x to find EG. ANSWER: EG = 6 + 5 = 11 33.5 ANSWER: 4. 11 SOLUTION: The diagonals of a rectangle are congruent and 6. If , bisect each other. So, is an isosceles triangle. find . Then, SOLUTION: By the Vertical Angle Theorem, All four angles of a rectangle are right angles. So,

Therefore,

ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle. ANSWER: 51

7. PROOF If ABDE is a rectangle and ,

5. If FD = 3x – 7 and EG = x + 5, find EG. prove that . SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.

Use the value of x to find EG. EG = 6 + 5 = 11 SOLUTION: You need to walk through the proof step by step. ANSWER: Look over what you are given and what you need to 11 prove. You are given ABDE is a rectangle and . You need to prove . Use the 6. If , properties that you have learned about rectangles to walk through the proof. find .

SOLUTION: Given: ABDE is a rectangle; All four angles of a rectangle are right angles. So, Prove:

ANSWER: Statements(Reasons)

51 1. ABDE is a rectangle; . (Given)

7. PROOF If ABDE is a rectangle and , 2. ABDE is a parallelogram. (Def. of rectangle) prove that . 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER:

SOLUTION: Given: ABDE is a rectangle;

You need to walk through the proof step by step. Prove: Look over what you are given and what you need to prove. You are given ABDE is a rectangle and . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABDE is a rectangle; Statements(Reasons)

Prove: 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) Statements(Reasons) 6. (SAS) 1. ABDE is a rectangle; . 7. (CPCTC) (Given) 2. ABDE is a parallelogram. (Def. of COORDINATE GEOMETRY Graph each rectangle) quadrilateral with the given vertices. Determine 3. (Opp. sides of a .) whether the figure is a rectangle. Justify your answer using the indicated formula.

4. are right angles.(Def. of rectangle) 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula 5. (All rt .) SOLUTION: 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove:

Use the slope formula to find the slope of the sides of the quadrilateral. Statements(Reasons) 1. ABDE is a rectangle; . (Given) 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) There are no parallel sides, so WXYZ is not a 7. (CPCTC) parallelogram. Therefore, WXYZ is not a rectangle.

COORDINATE GEOMETRY Graph each ANSWER: quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your No; slope of , slope of , slope of answer using the indicated formula. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula , and slope of . Slope of SOLUTION: slope of , and slope of slope of , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

ANSWER:

No; slope of , slope of , slope of

, and slope of . Slope of First, Use the Distance formula to find the lengths of slope of , and slope of slope of the sides of the quadrilateral. , so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance ANSWER: Formula Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a SOLUTION: parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral. FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each measure.

so ABCD is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, ABCD is a rectangle. 10. BC

ANSWER: SOLUTION: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle. ANSWER: 2 ft

11. DB SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

FENCING X-braces are also used to provide support in rectangular fencing. If AB = 6 feet, AD = 2 feet, and , find each ANSWER: measure. 6.3 ft

12. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then,

10. BC SOLUTION: By the Vertical Angle Theorem, The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

ANSWER: ANSWER: 2 ft 50 13. 11. DB SOLUTION: SOLUTION: The diagonals of a rectangle are congruent and All four angles of a rectangle are right angles. So, bisect each other. So, is an isosceles is a right triangle. By the Pythagorean triangle. 2 2 2 Theorem, DB = BC + DC . Then, BC = AD = 2 All four angles of a rectangle are right angles. DC = AB = 6

ANSWER: ANSWER: 25 6.3 ft CCSS REGULARITY Quadrilateral WXYZ is a 12. rectangle. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, 14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: The opposite sides of a rectangle are parallel and By the Vertical Angle Theorem, congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 ANSWER: Use the value of x to find WX. 50 WX = 1 + 4 = 5.

13. ANSWER: SOLUTION: 5 The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. triangle. SOLUTION: Then, All four angles of a rectangle are right angles. The diagonals of a rectangle bisect each other. So,

PY = WP.

The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles triangle. 25 CCSS REGULARITY Quadrilateral WXYZ is a rectangle. ANSWER: 43

16. If , find .

14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 2x + 3 = x + 4 x = 1 Use the value of x to find WX. WX = 1 + 4 = 5.

ANSWER: ANSWER: 5 39 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: SOLUTION: The diagonals of a rectangle bisect each other. So, The diagonals of a rectangle are congruent and PY = WP. bisect each other. So, ZP = PY.

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38.

ANSWER: 38 ANSWER: 18. If , find 43 . 16. If , find SOLUTION: . All four angles of a rectangle are right angles. So, SOLUTION: All four angles of a rectangle are right angles. So,

ANSWER: 48

19. If , find ANSWER:

39 . SOLUTION: 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. All four angles of a rectangle are right angles. So, SOLUTION: is a right triangle. Then, The diagonals of a rectangle are congruent and bisect each other. So, ZP = PY.

Therefore, Since is a Then ZP = 4(7) – 9 = 19. Therefore, ZX = 2(ZP) = 38. transversal of parallel sides and , alternate interior angles WZX and ZXY are congruent. So, ANSWER: = = 46. 38 ANSWER: 18. If , find 46 . PROOF Write a two-column proof. SOLUTION: 20. Given: ABCD is a rectangle. All four angles of a rectangle are right angles. So, Prove:

SOLUTION: ANSWER: You need to walk through the proof step by step. 48 Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. 19. If , find You need to prove . Use the . properties that you have learned about rectangles to walk through the proof. SOLUTION: All four angles of a rectangle are right angles. So, Given: ABCD is a rectangle. is a right triangle. Then, Prove: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) Therefore, Since is a 4. (Refl. Prop.) transversal of parallel sides and , alternate 5. (Diagonals of a rectangle are .) interior angles WZX and ZXY are congruent. So, 6. (SSS) = = 46. ANSWER: ANSWER: Proof: 46 Statements (Reasons) 1. ABCD is a rectangle. (Given) PROOF Write a two-column proof. 2. ABCD is a parallelogram. (Def. of rectangle) Given: 20. ABCD is a rectangle. Prove: 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS)

21. Given: QTVW is a rectangle. SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof. Prove:

Given: ABCD is a rectangle. SOLUTION: Prove: You need to walk through the proof step by step. Proof: Look over what you are given and what you need to Statements (Reasons) prove. Here, you are given QTVW is a rectangle and 1. ABCD is a rectangle. (Given) . You need to prove . Use 2. ABCD is a parallelogram. (Def. of rectangle) the properties that you have learned about rectangles 3. (Opp. sides of a .) to walk through the proof. 4. (Refl. Prop.)

5. (Diagonals of a rectangle are .) Given: QTVW is a rectangle.

6. (SSS)

ANSWER: Proof: Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) Prove: 3. (Opp. sides of a .) Proof: Statements (Reasons) 4. (Refl. Prop.) 1. QTVW is a rectangle; .(Given) 5. (Diagonals of a rectangle are .) 2. QTVW is a parallelogram. (Def. of rectangle) 6. (SSS) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 21. Given: QTVW is a rectangle.

5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) Prove: 11. QS = RT (Subst.)

SOLUTION: 12. (Def. of segs.) You need to walk through the proof step by step. 13. (SAS) Look over what you are given and what you need to prove. Here, you are given QTVW is a rectangle and ANSWER: Proof: . You need to prove . Use Statements (Reasons) the properties that you have learned about rectangles to walk through the proof. 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) Given: QTVW is a rectangle. 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) Prove: 9. QR + RS = RS + ST (Add. prop.) Proof: 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) Statements (Reasons) 11. QS = RT (Subst.) 1. QTVW is a rectangle; .(Given) 12. (Def. of segs.) 2. QTVW is a parallelogram. (Def. of rectangle) 13. (SAS) 3. (Opp sides of a .) COORDINATE GEOMETRY Graph each

4. are right angles.(Def. of rectangle) quadrilateral with the given vertices. Determine 5. (All rt .) whether the figure is a rectangle. Justify your 6. QR = ST (Def. of segs.) answer using the indicated formula. 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) SOLUTION: 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER: Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) Use the slope formula to find the slope of the sides of the quadrilateral. 4. are right angles.(Def. of rectangle)

Yes; slope of slope of , slope

of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula

The slopes of each pair of opposite sides are equal. SOLUTION: So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral WXYZ is a parallelogram. The products of the slopes of the adjacent sides are –1. So, any two adjacent sides are perpendicular to each other. That is, all the four angles are right angles. Therefore, WXYZ is a rectangle.

ANSWER:

Yes; slope of slope of , slope

of slope of . So WXYZ is a parallelogram. The product of the slopes of consecutive sides is –1, so the consecutive sides are First, Use the Distance formula to find the lengths of perpendicular and form right angles. Thus, WXYZ is a the sides of the quadrilateral. rectangle.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle. 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula ANSWER: SOLUTION: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a rectangle.

First, Use the Distance formula to find the lengths of the sides of the quadrilateral.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION: so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER:

No; so JKLM is a First, use the Distance Formula to find the lengths of the sides of the quadrilateral. parallelogram; so

the diagonals are not congruent. Thus, JKLM is not a rectangle.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: 24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula Yes; so QRST is a SOLUTION: parallelogram. so the diagonals are congruent. QRST is a rectangle.

First, use the Distance Formula to find the lengths of the sides of the quadrilateral. 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a First, use the slope formula to find the lengths of the sides of the quadrilateral. parallelogram. so the diagonals are congruent. QRST is a rectangle.

The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. Therefore, the quadrilateral GHJK is a parallelogram. None of the adjacent sides have slopes whose 25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula product is –1. So, the angles are not right angles. SOLUTION: Therefore, GHJK is not a rectangle. ANSWER:

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are not perpendicular. Thus, GHJK is not a rectangle.

First, use the slope formula to find the lengths of the sides of the quadrilateral.

Quadrilateral ABCD is a rectangle. Find each measure if .

No; slope of slope of and slope of

slope of . So, GHJK is a parallelogram. The product of the slopes of consecutive sides ≠ –1, so the consecutive sides are ANSWER: not perpendicular. Thus, GHJK is not a rectangle. 50

27. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: 40 Quadrilateral ABCD is a rectangle. Find each 28. measure if . SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠3 = m∠2 = 40. ANSWER:

26. 40 SOLUTION: 29. All four angles of a rectangle are right angles. So, SOLUTION:

All four angles of a rectangle are right angles. So,

The measures of angles 2 and 3 are equal as they and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50.

are alternate interior angles. The sum of the three angles of a triangle is 180. Since the diagonals of a rectangle are congruent and Therefore, m∠5 = 180 – (50 + 50) = 80. bisect each other, the triangle with the angles 3, 7 ANSWER: and 8 is an isosceles triangle. So, m∠7 = m∠3. 80 Therefore, m∠7 = m∠2 = 40.

ANSWER: 30. 40 SOLUTION: All four angles of a rectangle are right angles. So, 28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. The measures of angles 1 and 4 are equal as they Therefore, m∠3 = m∠2 = 40. are alternate interior angles. Therefore, m∠4 = m∠1 = 50. ANSWER: Since the diagonals of a rectangle are congruent and 40 bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 =

29. m∠4 = 50. SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 50

31. SOLUTION: The measures of angles 1 and 4 are equal as they are alternate interior angles. The measures of angles 2 and 3 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m 3 = m 7 = 40. and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. ∠ ∠ The sum of the three angles of a triangle is 180. The sum of the three angles of a triangle is 180. Therefore, m 8 = 180 – (40 + 40) = 100. Therefore, m∠5 = 180 – (50 + 50) = 80. ∠ ANSWER: ANSWER: 80 100 CCSS MODELING 30. 32. Jody is building a new bookshelf using wood and metal supports like the one SOLUTION: shown. To what length should she cut the metal All four angles of a rectangle are right angles. So, supports in order for the bookshelf to be square, which means that the angles formed by the shelves and the vertical supports are all right angles? Explain your reasoning. The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 and 6 is an isosceles triangle. Therefore, m∠6 = m∠4 = 50.

ANSWER: SOLUTION: 50 In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we 31. know the length of the bookshelves and the distance SOLUTION: between them, use the Pythagorean Theorem to find

The measures of angles 2 and 3 are equal as they the lengths of the metal supports. are alternate interior angles. Let x be the length of each support. Express 3 feet

Since the diagonals of a rectangle are congruent and as 36 inches. bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. Therefore, m∠8 = 180 – (40 + 40) = 100. ANSWER: Therefore, the metal supports should each be 39 100 inches long or 3 feet 3 inches long.

32. CCSS MODELING Jody is building a new ANSWER: bookshelf using wood and metal supports like the one 3 ft 3 in.; Sample answer: In order for the bookshelf shown. To what length should she cut the metal to be square, the lengths of all of the metal supports supports in order for the bookshelf to be square, should be equal. Since I knew the length of the which means that the angles formed by the shelves bookshelves and the distance between them, I used and the vertical supports are all right angles? Explain the Pythagorean Theorem to find the lengths of the your reasoning. metal supports. PROOF Write a two-column proof. 33. Theorem 6.13 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove SOLUTION: . Use the properties that you have learned In order for the bookshelf to be square, the lengths of about rectangles to walk through the proof. all of the metal supports should be equal. Since we know the length of the bookshelves and the distance Given: WXYZ is a rectangle with diagonals between them, use the Pythagorean Theorem to find .

the lengths of the metal supports. Let x be the length of each support. Express 3 feet Prove: as 36 inches.

Proof: Therefore, the metal supports should each be 39 1. WXYZ is a rectangle with diagonals . inches long or 3 feet 3 inches long. (Given) 2. (Opp. sides of a .) ANSWER:

3 ft 3 in.; Sample answer: In order for the bookshelf 3. (Refl. Prop.) to be square, the lengths of all of the metal supports 4. are right angles. (Def. of should be equal. Since I knew the length of the rectangle) bookshelves and the distance between them, I used 5. (All right .) the Pythagorean Theorem to find the lengths of the 6. (SAS) metal supports. 7. (CPCTC) PROOF Write a two-column proof. 33. Theorem 6.13 ANSWER: Given: WXYZ is a rectangle with diagonals SOLUTION: . You need to walk through the proof step by step. Look over what you are given and what you need to Prove: prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Proof: Given: WXYZ is a rectangle with diagonals 1. WXYZ is a rectangle with diagonals . . (Given)

Prove: 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of rectangle) 5. (All right .) Proof: 6. (SAS) 1. WXYZ is a rectangle with diagonals . 7. (CPCTC) (Given) 2. (Opp. sides of a .) 34. Theorem 6.14 3. (Refl. Prop.) SOLUTION: 4. are right angles. (Def. of You need to walk through the proof step by step. rectangle) Look over what you are given and what you need to 5. (All right .) prove. Here, you are given

6. (SAS) . You need to 7. (CPCTC) prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk ANSWER: through the proof. Given: WXYZ is a rectangle with diagonals . Given: Prove: Prove: WXYZ is a rectangle.

Proof: Proof: 1. WXYZ is a rectangle with diagonals . 1. (Given) (Given) 2. (SSS) 2. (Opp. sides of a .) 3. (CPCTC) 3. (Refl. Prop.) 4. (Def. of ) 4. are right angles. (Def. of 5. WXYZ is a parallelogram. (If both pairs of opp. rectangle) sides are , then quad. is .) 5. (All right .) 6. are supplementary. (Cons. 6. (SAS) are suppl.) 7. (CPCTC) 7. (Def of suppl.) 8. are right angles. (If 34. Theorem 6.14 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 SOLUTION: rt. , it has 4 rt. .) You need to walk through the proof step by step. 10. WXYZ is a rectangle. (Def. of rectangle) Look over what you are given and what you need to prove. Here, you are given ANSWER: . You need to Given: prove that WXYZ is a rectangle.. Use the properties Prove: WXYZ is a rectangle. that you have learned about rectangles to walk through the proof.

Given: Prove: WXYZ is a rectangle.

Proof: 1. (Given) 2. (SSS) 3. (CPCTC)

Proof: 4. (Def. of ) 1. (Given) 5. WXYZ is a parallelogram. (If both pairs of opp.

2. (SSS) sides are , then quad. is .) 6. are supplementary. (Cons. 3. (CPCTC) are suppl.) 4. (Def. of )

5. WXYZ is a parallelogram. (If both pairs of opp. 7. (Def of suppl.) sides are , then quad. is .) 8. are right angles. (If 6. are supplementary. (Cons. 2 and suppl., each is a rt. .) are suppl.) 9. are right angles. (If a has 1 7. (Def of suppl.) rt. , it has 4 rt. .)

8. are right angles. (If 10. WXYZ is a rectangle. (Def. of rectangle) 2 and suppl., each is a rt. .) PROOF Write a paragraph proof of each 9. are right angles. (If a has 1 statement. rt. , it has 4 rt. .) 35. If a parallelogram has one right angle, then it is a 10. WXYZ is a rectangle. (Def. of rectangle) rectangle. SOLUTION: ANSWER: You need to walk through the proof step by step.

Given: Look over what you are given and what you need to Prove: WXYZ is a rectangle. prove. Here, you are given a parallelogram with one right angle. You need to prove that it is a rectangle. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

Proof: 1. (Given) 2. (SSS) 3. (CPCTC) 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. ABCD is a parallelogram, and is a right angle. sides are , then quad. is .) Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the 6. are supplementary. (Cons. definition of a rectangle, ABCD is a rectangle. are suppl.) 7. (Def of suppl.) ANSWER: 8. are right angles. (If 2 and suppl., each is a rt. .) 9. are right angles. (If a has 1 rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) ABCD is a parallelogram, and is a right angle. PROOF Write a paragraph proof of each Since ABCD is a parallelogram and has one right statement. angle, then it has four right angles. So by the 35. If a parallelogram has one right angle, then it is a definition of a rectangle, ABCD is a rectangle. rectangle. SOLUTION: 36. If a quadrilateral has four right angles, then it is a You need to walk through the proof step by step. rectangle. Look over what you are given and what you need to prove. Here, you are given a parallelogram with one SOLUTION: right angle. You need to prove that it is a rectangle. You need to walk through the proof step by step. Let ABCD be a parallelogram with one right angle. Look over what you are given and what you need to Then use the properties of parallelograms to walk prove. Here, you are given a quadrilateral with four through the proof. right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. Then use the properties of parallelograms and rectangles to walk through the proof.

ANSWER:

ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle. ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a 36. If a quadrilateral has four right angles, then it is a rectangle, ABCD is a rectangle. rectangle. CONSTRUCTION SOLUTION: 37. Construct a rectangle using the construction for congruent segments and the You need to walk through the proof step by step. construction for a line perpendicular to another line Look over what you are given and what you need to through a point on the line. Justify each step of the prove. Here, you are given a quadrilateral with four construction. right angles. You need to prove that it is a rectangle. Let ABCD be a quadrilateral with four right angles. SOLUTION: Then use the properties of parallelograms and Sample answer: rectangles to walk through the proof. Step 1: Graph a line and place points P and Q on the line.

Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and right of P that intersects .

ABCD is a quadrilateral with four right angles. Step 3: Open the compass to a setting greater than ABCD is a parallelogram because both pairs of the distance from P to either point of intersection. opposite angles are congruent. By definition of a Place the compass on each point of intersection and rectangle, ABCD is a rectangle. draw arcs that intersect above . ANSWER: Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3.

Step 5: Repeat steps 2 and 3 using point Q.

Step 6: Draw line b through point Q and the ABCD is a quadrilateral with four right angles. intersection of the arcs drawn in step 5. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a Step 7: Using any compass setting, put the compass rectangle, ABCD is a rectangle. at P and draw an arc above it that intersects line a. Label the point of intersection R. 37. CONSTRUCTION Construct a rectangle using the construction for congruent segments and the Step 8: Using the same compass setting, put the construction for a line perpendicular to another line compass at Q and draw an arc above it that through a point on the line. Justify each step of the intersects line b. Label the point of intersection S. construction. SOLUTION: Step 9: Draw . Sample answer: Step 1: Graph a line and place points P and Q on the line.

Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and right of P that intersects .

Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Place the compass on each point of intersection and Since and , the measure of draw arcs that intersect above . angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 4: Draw line a through point P and the intersection of the arcs drawn in step 3. . The same compass setting was used to locate points R and S, so . If one pair of Step 5: Repeat steps 2 and 3 using point Q. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. Step 6: Draw line b through point Q and the A parallelogram with right angles is a rectangle. intersection of the arcs drawn in step 5. Thus, PRSQ is a rectangle.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. ANSWER: Label the point of intersection R. Sample answer: Since and , the Step 8: Using the same compass setting, put the measure of angles P and Q is 90 degrees. Lines that compass at Q and draw an arc above it that are perpendicular to the same line are parallel, so intersects line b. Label the point of intersection S. . The same compass setting was used to Step 9: Draw . locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

Since and , the measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so . The same compass setting was used to 38. SPORTS The end zone of a football field is 160 feet locate points R and S, so . If one pair of wide and 30 feet long. Kyle is responsible for opposite sides of a quadrilateral is both parallel and painting the field. He has finished the end zone. congruent, then the quadrilateral is a parallelogram. Explain how Kyle can confirm that the end zone is A parallelogram with right angles is a rectangle. the regulation size and be sure that it is also a Thus, PRSQ is a rectangle. rectangle using only a tape measure.

SOLUTION: ANSWER: Sample answer: If the diagonals of a parallelogram are congruent, then it is a rectangle. He should measure the diagonals of the end zone and both Sample answer: Since and , the pairs of opposite sides. If both pairs of opposite sides measure of angles P and Q is 90 degrees. Lines that are congruent, then the quadrilateral is a are perpendicular to the same line are parallel, so parallelogram. If the diagonals are congruent, then . The same compass setting was used to the end zone is a rectangle. locate points R and S, so . If one pair of ANSWER: opposite sides of a quadrilateral is both parallel and Sample answer: He should measure the diagonals of congruent, then the quadrilateral is a parallelogram. the end zone and both pairs of opposite sides. If the A parallelogram with right angles is a rectangle. diagonals and both pairs of opposite sides are Thus, PRSQ is a rectangle. congruent, then the end zone is a rectangle.

ALGEBRA Quadrilateral WXYZ is a rectangle.

39. If XW = 3, WZ = 4, and XZ = b, find YW. SOLUTION: The diagonals of a rectangle are congruent to each other. So, YW = XZ = b. All four angles of a rectangle are right angles. So, is a right 38. SPORTS The end zone of a football field is 160 feet 2 2 wide and 30 feet long. Kyle is responsible for triangle. By the Pythagorean Theorem, XZ = XW + 2 painting the field. He has finished the end zone. WZ . Explain how Kyle can confirm that the end zone is the regulation size and be sure that it is also a rectangle using only a tape measure. Therefore, YW = 5. SOLUTION: Sample answer: If the diagonals of a parallelogram ANSWER: are congruent, then it is a rectangle. He should 5 measure the diagonals of the end zone and both pairs of opposite sides. If both pairs of opposite sides are congruent, then the quadrilateral is a 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. parallelogram. If the diagonals are congruent, then SOLUTION: the end zone is a rectangle. The diagonals of a rectangle are congruent to each ANSWER: other. So, WY = XZ = 2c. All four angles of a rectangle are right angles. So, is a right Sample answer: He should measure the diagonals of 2 2 the end zone and both pairs of opposite sides. If the triangle. By the Pythagorean Theorem, XZ = XY + 2 diagonals and both pairs of opposite sides are ZY . congruent, then the end zone is a rectangle. ALGEBRA Quadrilateral WXYZ is a rectangle.

Therefore, WY = 2(5) = 10.

39. If XW = 3, WZ = 4, and XZ = b, find YW. ANSWER: SOLUTION: 10 The diagonals of a rectangle are congruent to each SIGNS other. So, YW = XZ = b. All four angles of a 41. The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko rectangle are right angles. So, is a right 2 2 be sure that the sign is a rectangle? Explain your triangle. By the Pythagorean Theorem, XZ = XW + reasoning. 2 WZ .

Therefore, YW = 5.

ANSWER: 5 SOLUTION: 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that The diagonals of a rectangle are congruent to each can be used to prove that it is a rectangle. If the other. So, WY = XZ = 2c. All four angles of a diagonals could be proven to be congruent, or if the rectangle are right angles. So, is a right angles could be proven to be right angles, then we triangle. By the Pythagorean Theorem, XZ2 = XY2 + could prove that the sign was a rectangle. 2 ZY . ANSWER:

No; sample answer: Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

PROOF Write a coordinate proof of each Therefore, WY = 2(5) = 10. statement. ANSWER: 42. The diagonals of a rectangle are congruent. 10 SOLUTION: Begin by positioning quadrilateral ABCD on a SIGNS 41. The sign below is in the foyer of Nyoko’s coordinate plane. Place vertex D at the origin. Let school. Based on the dimensions given, can Nyoko the length of the bases be a units and the height be b be sure that the sign is a rectangle? Explain your units. Then the rest of the vertices are A(0, b), B(a,

reasoning. b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD is a rectangle. Prove: SOLUTION:

Both pairs of opposite sides are congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle. If the diagonals could be proven to be congruent, or if the angles could be proven to be right angles, then we could prove that the sign was a rectangle. Proof: ANSWER: Use the Distance Formula to find the lengths of the No; sample answer: Both pairs of opposite sides are diagonals. congruent, so the sign is a parallelogram, but no measure is given that can be used to prove that it is a rectangle.

PROOF Write a coordinate proof of each statement. The diagonals, have the same length, so 42. The diagonals of a rectangle are congruent. they are congruent. SOLUTION: ANSWER: Begin by positioning quadrilateral ABCD on a Given: ABCD is a rectangle. coordinate plane. Place vertex D at the origin. Let Prove: the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle and you need to prove . Use the properties that you have learned about rectangles to walk through the proof. Proof: Given: ABCD is a rectangle. Use the Distance Formula to find Prove: and . have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION:

Proof: Begin by positioning parallelogram ABCD on a Use the Distance Formula to find the lengths of the coordinate plane. Place vertex A at the origin. Let diagonals. the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + a, c), and D(b, c). You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given The diagonals, have the same length, so ABCD and and you need to prove that they are congruent. ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the ANSWER: proof. Given: ABCD is a rectangle. Prove: Given: ABCD and Prove: ABCD is a rectangle.

Proof: Proof: Use the Distance Formula to find and . have the same length, so they are

congruent. AC = BD. So,

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. SOLUTION: Begin by positioning parallelogram ABCD on a coordinate plane. Place vertex A at the origin. Let the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + Because A and B are different points, a, c), and D(b, c). You need to walk through the a ≠ 0. Then b = 0. The slope of is undefined proof step by step. Look over what you are given and what you need to prove. Here, you are given and the slope of . Thus, . ∠DAB is ABCD and and you need to prove that a right angle and ABCD is a rectangle. ABCD is a rectangle. Use the properties that you ANSWER: have learned about rectangles to walk through the

proof. Given: ABCD and Prove: ABCD is a rectangle. Given: ABCD and Prove: ABCD is a rectangle.

Proof:

But AC = BD. So,

AC = BD. So,

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined Because A and B are different points, and the slope of . Thus, . ∠DAB is a ≠ 0. Then b = 0. The slope of is undefined a right angle and ABCD is a rectangle.. and the slope of . Thus, . ∠DAB is 44. MULTIPLE REPRESENTATIONS In the a right angle and ABCD is a rectangle. problem, you will explore properties of other special parallelograms. ANSWER: a. GEOMETRIC Draw three parallelograms, each Given: ABCD and with all four sides congruent. Label one Prove: ABCD is a rectangle. parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

Proof: c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

But AC = BD. So,

Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle.. b. Use a protractor to measure each angle listed in the table. 44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. c. a. GEOMETRIC Draw three parallelograms, each Sample answer: Each of the angles listed in the with all four sides congruent. Label one table are right angles. These angles were each parallelogram ABCD, one MNOP, and one WXYZ. formed by diagonals. The diagonals of a Draw the two diagonals of each parallelogram and parallelogram with four congruent sides are

label the intersections R. perpendicular. b. TABULAR Use a protractor to measure the ANSWER: appropriate angles and complete the table below. a. Sample answer:

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, b. Use a protractor to measure each angle listed in the table. Find the values of x and y.

c. Sample answer: Each of the angles listed in the table are right angles. These angles were each formed by diagonals. The diagonals of a parallelogram with four congruent sides are perpendicular.

ANSWER: a. Sample answer: SOLUTION: All four angles of a rectangle are right angles. So,

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So,

The sum of the angles of a triangle is 180.

So, c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular. By the Vertical Angle Theorem,

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

ANSWER: x = 6, y = −10

46. CCSS CRITIQUE Parker says that any two congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: SOLUTION: All four angles of a rectangle are right angles. So, When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So, In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

So, Tamika is correct.

ANSWER: Tamika; Sample answer: When two congruent ANSWER: triangles are arranged to form a quadrilateral, two of x = 6, y = −10 the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of 46. CCSS CRITIQUE Parker says that any two the angles in the congruent triangles has to be a right congruent acute triangles can be arranged to make a angle. rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is 47. REASONING In the diagram at the right, lines n, p, either of them correct? Explain your reasoning. q,and r are parallel and lines l and m are parallel. How many rectangles are formed by the intersecting SOLUTION: lines? When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

In order for the quadrilateral to be a rectangle, one of SOLUTION: the angles in the congruent triangles has to be a right Let the lines n, p, q, and r intersect the line l at the angle. points A, B, C, and D respectively. Similarly, let the lines intersect the line m at the points W, X, Y, and Z.

So, Tamika is correct.

ANSWER: Tamika; Sample answer: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. In order for the quadrilateral to be a rectangle, one of Then the rectangles formed are ABXW, ACYW, the angles in the congruent triangles has to be a right BCYX, BDZX, CDZY, and ADZW. Therefore, 6 angle. rectangles are formed by the intersecting lines.

47. REASONING In the diagram at the right, lines n, p, ANSWER: q,and r are parallel and lines l and m are parallel. 6 How many rectangles are formed by the intersecting lines? 48. OPEN ENDED Write the equations of four lines having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other SOLUTION: and perpendicular to lines parallel to the y-axis. Let the lines n, p, q, and r intersect the line l at the Selecting y = 0 and x = 0 will form 2 sides of the points A, B, C, and D respectively. Similarly, let the rectangle. The lines y = 4 and x = 6 form the other 2

lines intersect the line m at the points W, X, Y, and Z. sides.

x = 0, x = 6, y = 0, y = 4;

Then the rectangles formed are ABXW, ACYW, The length of is 6 − 0 or 6 units and the length of BCYX, BDZX, CDZY, and ADZW. Therefore, 6 is 6 – 0 or 6 units. The slope of AB is 0 and the rectangles are formed by the intersecting lines. slope of DC is 0. Since a pair of sides of the quadrilateral is both parallel and congruent, by ANSWER: Theorem 6.12, the quadrilateral is a parallelogram. 6 Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they 48. OPEN ENDED Write the equations of four lines form is 90. By Theorem 6.6, if a parallelogram has having intersections that form the vertices of a one right angle, it has four right angles. Therefore, by rectangle. Verify your answer using coordinate definition, the parallelogram is a rectangle. geometry. SOLUTION: ANSWER: Sample answer: Sample answer: x = 0, x = 6, y = 0, A rectangle has 4 right angles and each pair of y = 4; opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

x = 0, x = 6, y = 0, y = 4;

definition, the parallelogram is a rectangle. definition of rectangle, all rectangles have both pairs ANSWER: of opposite sides parallel. So rectangles are always parallelograms. Sample answer: x = 0, x = 6, y = 0,

y = 4; Parallelograms with right angles are rectangles, so some parallelograms are rectangles, but others with non-right angles are not.

ANSWER: Sample answer: All rectangles are parallelograms because, by definition, both pairs of opposite sides are parallel. Parallelograms with right angles are rectangles, so some parallelograms are rectangles,

but others with non-right angles are not. The length of is 6 − 0 or 6 units and the length of 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = is 6 – 0 or 6 units. The slope of AB is 0 and the 13, what values of x and y make parallelogram slope of DC is 0. Since a pair of sides of the FGHJ a rectangle? quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by A x = 3, y = 4 definition, the parallelogram is a rectangle. B x = 4, y = 3 C x = 7, y = 8 49. WRITING IN MATH Explain why all rectangles D x = 8, y = 7 are parallelograms, but all parallelograms are not rectangles. SOLUTION: The opposite sides of a parallelogram are congruent. SOLUTION: So, FJ = GH. Sample answer: A parallelogram is a quadrilateral in –3x + 5y = 11 FJ = GH which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs If the diagonals of a parallelogram are congruent, of opposite sides parallel. So rectangles are always then it is a rectangle. Since FGHJ is a parallelogram, parallelograms. the diagonals bisect each other. So, if it is a rectangle, then will be an isosceles triangle Parallelograms with right angles are rectangles, so with That is, 3x + y = 13. some parallelograms are rectangles, but others with non-right angles are not. Add the two equations to eliminate the x-term and ANSWER: solve for y. –3x + 5y = 11 Sample answer: All rectangles are parallelograms 3x + y = 13 because, by definition, both pairs of opposite sides 6y = 24 Add to eliminate x-term. are parallel. Parallelograms with right angles are rectangles, so some parallelograms are rectangles, y = 4 Divide each side by 6. but others with non-right angles are not. Use the value of y to find the value of x. 50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 3x + 4 = 13 Substitute. 13, what values of x and y make parallelogram 3x = 9 Subtract 4 from each side. FGHJ a rectangle? x = 3 Divide each side by 3.

So, x = 3, y = 4, the correct choice is A.

ANSWER: A A x = 3, y = 4 B x = 4, y = 3 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 C x = 7, y = 8 feet longer than the other. Which of the following D x = 8, y = 7 equations could be used to find r, the shorter side of SOLUTION: the playground? The opposite sides of a parallelogram are congruent. F 10r + r = 80 So, FJ = GH. G 4r + 10 = 80 –3x + 5y = 11 FJ = GH H r(r + 10) = 80 J 2(r + 10) + 2r = 80 If the diagonals of a parallelogram are congruent, then it is a rectangle. Since FGHJ is a parallelogram, SOLUTION: the diagonals bisect each other. So, if it is a The formula to find the perimeter of a rectangle is rectangle, then will be an isosceles triangle given by P = 2(l + w). The perimeter is given to be 8 with That is, 3x + y = 13. ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. Therefore, the correct choice is J. Add the two equations to eliminate the x-term and solve for y. ANSWER: –3x + 5y = 11 J 3x + y = 13

6y = 24 Add to eliminate x-term. 52. SHORT RESPONSE What is the measure of

y = 4 Divide each side by 6. ?

Use the value of y to find the value of x. 3x + 4 = 13 Substitute. 3x = 9 Subtract 4 from each side. x = 3 Divide each side by 3.

So, x = 3, y = 4, the correct choice is A. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and A bisect each other. So, is an isosceles triangle

51. ALGEBRA A rectangular playground is surrounded with by an 80-foot fence. One side of the playground is 10 So, feet longer than the other. Which of the following By the Exterior Angle Theorem, equations could be used to find r, the shorter side of the playground? ANSWER: F 10r + r = 80 112 G 4r + 10 = 80 H r(r + 10) = 80 53. SAT/ACT If p is odd, which of the following must J 2(r + 10) + 2r = 80 also be odd? SOLUTION: A 2p The formula to find the perimeter of a rectangle is B 2p + 2

given by P = 2(l + w). The perimeter is given to be 8 C ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. D Therefore, the correct choice is J. E p + 2 ANSWER: SOLUTION: J Analyze each answer choice to determine which expression is odd when p is odd. 52. SHORT RESPONSE What is the measure of Choice A 2p ? This is a multiple of 2 so it is even. Choice B 2p + 2 This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

Choice C

If p is odd , is not a whole number. SOLUTION: Choice D 2p – 2 The diagonals of a rectangle are congruent and This can be rewritten as 2(p - 1) which is also a bisect each other. So, is an isosceles triangle multiple of 2 so it is even. with Choice E p + 2 So, Adding 2 to an odd number results in an odd By the Exterior Angle Theorem, number. This is odd. The correct choice is E.

6-4 RectanglesANSWER: ANSWER: 112 E

53. SAT/ACT If p is odd, which of the following must ALGEBRA Find x and y so that the also be odd? quadrilateral is a parallelogram. A 2p B 2p + 2

C D 54. E p + 2 SOLUTION: SOLUTION: The opposite sides of a parallelogram are congruent. Analyze each answer choice to determine which expression is odd when p is odd. Choice A 2p This is a multiple of 2 so it is even. In a parallelogram, consecutive interior angles are Choice B 2p + 2 supplementary. This can be rewritten as 2(p + 1) which is also a multiple of 2 so it is even.

Choice C If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a ANSWER: multiple of 2 so it is even. x = 2, y = 41 Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. The correct choice is E.

ANSWER: E 55.

ALGEBRA Find x and y so that the SOLUTION: quadrilateral is a parallelogram. The opposite sides of a parallelogram are congruent.

The opposite sides of a parallelogram are parallel to 54. each other. So, the angles with the measures (3y + 3)° and (4y – 19)° are alternate interior angles and SOLUTION: hence they are equal. The opposite sides of a parallelogram are congruent.

In a parallelogram, consecutive interior angles are ANSWER: supplementary. x = 8, y = 22

eSolutions Manual - Powered by Cognero Page 21 56. ANSWER: SOLUTION: x = 2, y = 41 The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 Use the value of y in the second equation and solve for x. 55. 7 + 3 = 2x + 6 4 = 2x SOLUTION: 2 = x The opposite sides of a parallelogram are congruent. ANSWER: x = 2, y = 7

The opposite sides of a parallelogram are parallel to 57. COORDINATE GEOMETRY Find the each other. So, the angles with the measures (3y + coordinates of the intersection of the diagonals of 3)° and (4y – 19)° are alternate interior angles and parallelogram ABCD with vertices A(1, 3), B(6, 2), C hence they are equal. (4, –2), and D(–1, –1). SOLUTION: The diagonals of a parallelogram bisect each other, so the intersection is the midpoint of either diagonal. Find the midpoint of or . ANSWER: x = 8, y = 22

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: 56. (2.5, 0.5)

SOLUTION: Refer to the figure. The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6. Solve the first equation to find the value of y. 2y = 14 y = 7 Use the value of y in the second equation and solve for x.

7 + 3 = 2x + 6 4 = 2x 2 = x 58. If , name two congruent angles. SOLUTION: ANSWER: x = 2, y = 7 Since is an isosceles triangle. So,

57. COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of ANSWER: parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: 59. If , name two congruent segments. The diagonals of a parallelogram bisect each other, SOLUTION: so the intersection is the midpoint of either diagonal. In a triangle with two congruent angles, the sides Find the midpoint of or . opposite to the congruent angles are also congruent. So, .

ANSWER: Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

ANSWER: 60. If , name two congruent segments. (2.5, 0.5) SOLUTION: Refer to the figure. In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

58. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

60. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER:

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps to prevent the door from warping over time. If feet, PS = 7 feet, and , find each measure.

1. QR SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, QR = PS = 7 ft.

ANSWER: 7 ft

2. SQ SOLUTION: The diagonals of a rectangle bisect each other. So,

FARMING An X-brace on a rectangular barn door is both decorative and functional. It helps ANSWER: to prevent the door from warping over time.

If feet, PS = 7 feet, and

, find each measure. 3. SOLUTION: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. Then, By the Exterior Angle Theorem,

Therefore,

ANSWER: 1. QR 33.5 SOLUTION: The opposite sides of a rectangle are parallel and 4.

congruent. Therefore, QR = PS = 7 ft. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 7 ft bisect each other. So, is an isosceles triangle. Then, By the Vertical Angle Theorem, 2. SQ

SOLUTION: Therefore, The diagonals of a rectangle bisect each other. So,

ANSWER: 56.5 ALGEBRA Quadrilateral DEFG is a rectangle.

ANSWER: 5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each 3. other. So, FD = EG.

Therefore, ANSWER: ANSWER: 51 56.5 7. PROOF If ABDE is a rectangle and , ALGEBRA Quadrilateral DEFG is a rectangle. prove that .

5. If FD = 3x – 7 and EG = x + 5, find EG. SOLUTION: The diagonals of a rectangle are congruent to each SOLUTION: other. So, FD = EG. You need to walk through the proof step by step.

Look over what you are given and what you need to prove. You are given ABDE is a rectangle and . You need to prove . Use the Use the value of x to find EG. properties that you have learned about rectangles to EG = 6 + 5 = 11 walk through the proof.

ANSWER:

11 Given: ABDE is a rectangle; Prove: 6. If , find . SOLUTION: All four angles of a rectangle are right angles. So,

prove that . 7. (CPCTC) ANSWER: Given: ABDE is a rectangle; Prove:

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your Statements(Reasons) answer using the indicated formula. 1. ABDE is a rectangle; . 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula (Given) SOLUTION: 2. ABDE is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. (SAS) 7. (CPCTC)

ANSWER: Given: ABDE is a rectangle; Prove: Use the slope formula to find the slope of the sides of the quadrilateral.

7. (CPCTC) , and slope of . Slope of

COORDINATE GEOMETRY Graph each slope of , and slope of slope of quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your , so WXYZ is not a parallelogram. Therefore, answer using the indicated formula. WXYZ is not a rectangle. 8. W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula SOLUTION:

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula Use the slope formula to find the slope of the sides of the quadrilateral. SOLUTION:

There are no parallel sides, so WXYZ is not a parallelogram. Therefore, WXYZ is not a rectangle. First, Use the Distance formula to find the lengths of ANSWER: the sides of the quadrilateral. No; slope of , slope of , slope of

, and slope of . Slope of

So the diagonals are congruent. Thus, ABCD is a rectangle.

ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a parallelogram. , so the diagonals are congruent. Thus, ABCD is a rectangle.

9. A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance Formula SOLUTION:

so ABCD is a parallelogram.

A parallelogram is a rectangle if the diagonals are 10. BC congruent. Use the Distance formula to find the lengths of the diagonals. SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft.

So the diagonals are congruent. Thus, ABCD is a ANSWER: rectangle. 2 ft ANSWER: Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a 11. DB parallelogram. , so the diagonals are SOLUTION:

congruent. Thus, ABCD is a rectangle. All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . BC = AD = 2 DC = AB = 6

ANSWER: 6.3 ft

By the Vertical Angle Theorem,

ANSWER: 10. BC 50 SOLUTION: 13. The opposite sides of a rectangle are parallel and congruent. Therefore, BC = AD = 2 ft. SOLUTION: The diagonals of a rectangle are congruent and ANSWER: bisect each other. So, is an isosceles 2 ft triangle. Then, 11. DB All four angles of a rectangle are right angles. SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. By the Pythagorean 2 2 2 Theorem, DB = BC + DC . ANSWER: BC = AD = 2 25 DC = AB = 6 CCSS REGULARITY Quadrilateral WXYZ is a rectangle.

ANSWER: 6.3 ft

12. 14. If ZY = 2x + 3 and WX = x + 4, find WX. SOLUTION: The diagonals of a rectangle are congruent and SOLUTION: bisect each other. So, is an isosceles The opposite sides of a rectangle are parallel and triangle. congruent. So, ZY = WX. 2x + 3 = x + 4 Then, x = 1 Use the value of x to find WX. WX = 1 + 4 = 5.

By the Vertical Angle Theorem, ANSWER: 5 ANSWER: 50 15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION: 13. The diagonals of a rectangle bisect each other. So, SOLUTION: PY = WP. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. The diagonals of a rectangle are congruent and Then, bisect each other. So, is an isosceles triangle. All four angles of a rectangle are right angles.

ANSWER: ANSWER: 43 25 16. If , find CCSS REGULARITY Quadrilateral WXYZ is a . rectangle. SOLUTION: All four angles of a rectangle are right angles. So,

14. If ZY = 2x + 3 and WX = x + 4, find WX.

SOLUTION: The opposite sides of a rectangle are parallel and congruent. So, ZY = WX. 2x + 3 = x + 4 ANSWER: x = 1 39 Use the value of x to find WX. 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. WX = 1 + 4 = 5. SOLUTION: ANSWER: The diagonals of a rectangle are congruent and 5 bisect each other. So, ZP = PY.

15. If PY = 3x – 5 and WP = 2x + 11, find ZP. SOLUTION:

The diagonals of a rectangle bisect each other. So, Then ZP = 4(7) – 9 = 19.

PY = WP. Therefore, ZX = 2(ZP) = 38. ANSWER: 38 The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle. 18. If , find . SOLUTION: All four angles of a rectangle are right angles. So, ANSWER: 43

16. If , find . SOLUTION: ANSWER: All four angles of a rectangle are right angles. So, 48 19. If , find .

SOLUTION: All four angles of a rectangle are right angles. So, is a right triangle. Then, ANSWER: 39 17. If ZP = 4x – 9 and PY = 2x + 5, find ZX. SOLUTION: Therefore, Since is a The diagonals of a rectangle are congruent and transversal of parallel sides and , alternate

bisect each other. So, ZP = PY. interior angles WZX and ZXY are congruent. So,

= = 46.

ANSWER: Then ZP = 4(7) – 9 = 19. 46 Therefore, ZX = 2(ZP) = 38. PROOF Write a two-column proof. ANSWER: 20. Given: ABCD is a rectangle. 38 Prove:

18. If , find .

SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: You need to walk through the proof step by step.

Look over what you are given and what you need to prove. Here, you are given ABCD is a rectangle. You need to prove . Use the properties that you have learned about rectangles to walk through the proof. ANSWER: 48 Given: ABCD is a rectangle. Prove: 19. If , find Proof: . Statements (Reasons) 1. ABCD is a rectangle. (Given) SOLUTION: 2. ABCD is a parallelogram. (Def. of rectangle) All four angles of a rectangle are right angles. So, 3. (Opp. sides of a .) is a right triangle. Then, 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS)

PROOF Write a two-column proof. 6. (SSS) 20. Given: ABCD is a rectangle. Prove: 21. Given: QTVW is a rectangle.

Proof: Given: QTVW is a rectangle. Statements (Reasons) 1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp. sides of a .) 4. (Refl. Prop.) 5. (Diagonals of a rectangle are .) 6. (SSS) Prove: Proof: ANSWER: Statements (Reasons) Proof: 1. QTVW is a rectangle; .(Given)

Statements (Reasons) 2. QTVW is a parallelogram. (Def. of rectangle)

1. ABCD is a rectangle. (Given) 2. ABCD is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 3. (Opp. sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 4. (Refl. Prop.) 6. QR = ST (Def. of segs.) 5. (Diagonals of a rectangle are .)

6. (SSS) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) Given: 21. QTVW is a rectangle. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

ANSWER:

Prove: Proof: Statements (Reasons) SOLUTION: 1. QTVW is a rectangle; .(Given) You need to walk through the proof step by step. 2. QTVW is a parallelogram. (Def. of rectangle) Look over what you are given and what you need to 3. (Opp sides of a .) prove. Here, you are given QTVW is a rectangle and 4. are right angles.(Def. of rectangle) . You need to prove . Use 5. (All rt .) the properties that you have learned about rectangles to walk through the proof. 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) Given: QTVW is a rectangle. 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

Prove: COORDINATE GEOMETRY Graph each Proof: quadrilateral with the given vertices. Determine Statements (Reasons) whether the figure is a rectangle. Justify your 1. QTVW is a rectangle; .(Given) answer using the indicated formula. 2. QTVW is a parallelogram. (Def. of rectangle) 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula 3. (Opp sides of a .) SOLUTION: 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) 8. RS = RS (Def. of segs.) 9. QR + RS = RS + ST (Add. prop.) 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) 11. QS = RT (Subst.) 12. (Def. of segs.) 13. (SAS)

Use the slope formula to find the slope of the sides of ANSWER: the quadrilateral. Proof: Statements (Reasons) 1. QTVW is a rectangle; .(Given) 2. QTVW is a parallelogram. (Def. of rectangle) 3. (Opp sides of a .) 4. are right angles.(Def. of rectangle) 5. (All rt .) 6. QR = ST (Def. of segs.) 7. (Refl. Prop.) The slopes of each pair of opposite sides are equal. 8. RS = RS (Def. of segs.) So, the two pairs of opposite sides are parallel. 9. QR + RS = RS + ST (Add. prop.) Therefore, the quadrilateral WXYZ is a parallelogram. 10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.) The products of the slopes of the adjacent sides are 11. QS = RT (Subst.) –1. So, any two adjacent sides are perpendicular to 12. (Def. of segs.) each other. That is, all the four angles are right 13. (SAS) angles. Therefore, WXYZ is a rectangle.

COORDINATE GEOMETRY Graph each ANSWER: quadrilateral with the given vertices. Determine Yes; slope of slope of , slope whether the figure is a rectangle. Justify your answer using the indicated formula. of slope of . So WXYZ is a 22. W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula parallelogram. The product of the slopes of SOLUTION: consecutive sides is –1, so the consecutive sides are perpendicular and form right angles. Thus, WXYZ is a rectangle.

Use the slope formula to find the slope of the sides of the quadrilateral.

23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance Formula SOLUTION:

ANSWER: First, Use the Distance formula to find the lengths of the sides of the quadrilateral. Yes; slope of slope of , slope

So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a parallelogram; so the diagonals are not congruent. Thus, JKLM is not a 23. J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance rectangle. Formula SOLUTION:

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula

First, Use the Distance formula to find the lengths of SOLUTION: the sides of the quadrilateral.

so JKLM is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance formula to find the lengths of the diagonals.

First, use the Distance Formula to find the lengths of the sides of the quadrilateral. So the diagonals are not congruent. Thus, JKLM is not a rectangle.

ANSWER: No; so JKLM is a so QRST is a parallelogram; so parallelogram. the diagonals are not congruent. Thus, JKLM is not a A parallelogram is a rectangle if the diagonals are rectangle. congruent. Use the Distance Formula to find the lengths of the diagonals.

So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

24. Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula SOLUTION:

25. G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula SOLUTION:

First, use the Distance Formula to find the lengths of the sides of the quadrilateral.

so QRST is a parallelogram. A parallelogram is a rectangle if the diagonals are congruent. Use the Distance Formula to find the lengths of the diagonals.

First, use the slope formula to find the lengths of the sides of the quadrilateral. So the diagonals are congruent. Thus, QRST is a rectangle.

ANSWER: Yes; so QRST is a parallelogram. so the diagonals are congruent. QRST is a rectangle.

ANSWER:

First, use the slope formula to find the lengths of the sides of the quadrilateral.

Quadrilateral ABCD is a rectangle. Find each measure if .

26. The slopes of each pair of opposite sides are equal. So, the two pairs of opposite sides are parallel. SOLUTION: Therefore, the quadrilateral GHJK is a All four angles of a rectangle are right angles. So, parallelogram. None of the adjacent sides have slopes whose product is –1. So, the angles are not right angles. Therefore, GHJK is not a rectangle. ANSWER: ANSWER: 50 No; slope of slope of and slope of 27. slope of . So, GHJK is a parallelogram. The product of the slopes of SOLUTION: consecutive sides ≠ –1, so the consecutive sides are The measures of angles 2 and 3 are equal as they not perpendicular. Thus, GHJK is not a rectangle. are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: 40

28. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles.

Therefore, m∠3 = m∠2 = 40. Quadrilateral ABCD is a rectangle. Find each ANSWER: measure if . 40

29. SOLUTION: All four angles of a rectangle are right angles. So,

26.

SOLUTION: All four angles of a rectangle are right angles. So, The measures of angles 1 and 4 are equal as they are alternate interior angles. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 ANSWER: and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. 50 The sum of the three angles of a triangle is 180. Therefore, m∠5 = 180 – (50 + 50) = 80. 27. ANSWER: SOLUTION: 80 The measures of angles 2 and 3 are equal as they 30. are alternate interior angles. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 3, 7 All four angles of a rectangle are right angles. So, and 8 is an isosceles triangle. So, m∠7 = m∠3. Therefore, m∠7 = m∠2 = 40. ANSWER: 40 The measures of angles 1 and 4 are equal as they are alternate interior angles. 28. Therefore, m∠4 = m∠1 = 50. Since the diagonals of a rectangle are congruent and SOLUTION: bisect each other, the triangle with the angles 4, 5 The measures of angles 2 and 3 are equal as they and 6 is an isosceles triangle. Therefore, m∠6 = are alternate interior angles. m∠4 = 50. Therefore, m∠3 = m∠2 = 40. ANSWER: ANSWER: 50 40 31. 29. SOLUTION: SOLUTION: The measures of angles 2 and 3 are equal as they All four angles of a rectangle are right angles. So, are alternate interior angles. Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 3, 7 and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The measures of angles 1 and 4 are equal as they The sum of the three angles of a triangle is 180. are alternate interior angles. Therefore, m∠8 = 180 – (40 + 40) = 100. Therefore, m∠4 = m∠1 = 50. ANSWER: Since the diagonals of a rectangle are congruent and bisect each other, the triangle with the angles 4, 5 100 and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50. 32. CCSS MODELING Jody is building a new The sum of the three angles of a triangle is 180. bookshelf using wood and metal supports like the one

Therefore, m∠5 = 180 – (50 + 50) = 80. shown. To what length should she cut the metal ANSWER: supports in order for the bookshelf to be square, which means that the angles formed by the shelves 80 and the vertical supports are all right angles? Explain your reasoning. 30. SOLUTION: All four angles of a rectangle are right angles. So,

31. SOLUTION: The measures of angles 2 and 3 are equal as they are alternate interior angles. Since the diagonals of a rectangle are congruent and Therefore, the metal supports should each be 39 bisect each other, the triangle with the angles 3, 7 inches long or 3 feet 3 inches long. and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40. The sum of the three angles of a triangle is 180. ANSWER: Therefore, m∠8 = 180 – (40 + 40) = 100. 3 ft 3 in.; Sample answer: In order for the bookshelf to be square, the lengths of all of the metal supports ANSWER: should be equal. Since I knew the length of the 100 bookshelves and the distance between them, I used the Pythagorean Theorem to find the lengths of the 32. CCSS MODELING Jody is building a new metal supports. bookshelf using wood and metal supports like the one shown. To what length should she cut the metal PROOF Write a two-column proof. supports in order for the bookshelf to be square, 33. Theorem 6.13 which means that the angles formed by the shelves SOLUTION: and the vertical supports are all right angles? Explain You need to walk through the proof step by step. your reasoning. Look over what you are given and what you need to prove. Here, you are given WXYZ is a rectangle with diagonals . You need to prove . Use the properties that you have learned about rectangles to walk through the proof.

Given: WXYZ is a rectangle with diagonals . SOLUTION: Prove: In order for the bookshelf to be square, the lengths of all of the metal supports should be equal. Since we know the length of the bookshelves and the distance between them, use the Pythagorean Theorem to find the lengths of the metal supports. Let x be the length of each support. Express 3 feet Proof: as 36 inches. 1. WXYZ is a rectangle with diagonals . (Given) 2. (Opp. sides of a .) 3. (Refl. Prop.) 4. are right angles. (Def. of Therefore, the metal supports should each be 39 rectangle) inches long or 3 feet 3 inches long. 5. (All right .) 6. (SAS) ANSWER:

3 ft 3 in.; Sample answer: In order for the bookshelf 7. (CPCTC) to be square, the lengths of all of the metal supports should be equal. Since I knew the length of the ANSWER: bookshelves and the distance between them, I used Given: WXYZ is a rectangle with diagonals the Pythagorean Theorem to find the lengths of the . metal supports. Prove: PROOF Write a two-column proof. 33. Theorem 6.13 SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to Proof: prove. Here, you are given WXYZ is a rectangle with 1. WXYZ is a rectangle with diagonals . diagonals . You need to prove (Given) . Use the properties that you have learned 2. (Opp. sides of a .) about rectangles to walk through the proof. 3. (Refl. Prop.)

Given: WXYZ is a rectangle with diagonals 4. are right angles. (Def. of rectangle) . 5. (All right .)

Prove: 6. (SAS) 7. (CPCTC)

34. Theorem 6.14

Proof: SOLUTION: 1. WXYZ is a rectangle with diagonals . You need to walk through the proof step by step. Look over what you are given and what you need to (Given) prove. Here, you are given 2. (Opp. sides of a .) . You need to 3. (Refl. Prop.) prove that WXYZ is a rectangle.. Use the properties 4. are right angles. (Def. of that you have learned about rectangles to walk rectangle) through the proof. 5. (All right .)

6. (SAS) Given: Prove: WXYZ is a rectangle. 7. (CPCTC)

ANSWER: Given: WXYZ is a rectangle with diagonals .

7. (CPCTC) 10. WXYZ is a rectangle. (Def. of rectangle) ANSWER: 34. Theorem 6.14 Given: SOLUTION: Prove: WXYZ is a rectangle. You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here, you are given . You need to prove that WXYZ is a rectangle.. Use the properties that you have learned about rectangles to walk Proof: through the proof. 1. (Given) 2. (SSS)

Given: 3. (CPCTC)

Prove: WXYZ is a rectangle. 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.)

Proof: 7. (Def of suppl.) 1. (Given) 8. are right angles. (If

2. (SSS) 2 and suppl., each is a rt. .) 3. (CPCTC) 9. are right angles. (If a has 1

4. (Def. of ) rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) PROOF Write a paragraph proof of each 6. are supplementary. (Cons. statement. are suppl.) 35. If a parallelogram has one right angle, then it is a

7. (Def of suppl.) rectangle. 8. are right angles. (If SOLUTION: 2 and suppl., each is a rt. .) You need to walk through the proof step by step. 9. are right angles. (If a has 1 Look over what you are given and what you need to prove. Here, you are given a parallelogram with one

rt. , it has 4 rt. .) right angle. You need to prove that it is a rectangle.

10. WXYZ is a rectangle. (Def. of rectangle) Let ABCD be a parallelogram with one right angle. ANSWER: Then use the properties of parallelograms to walk through the proof. Given: Prove: WXYZ is a rectangle.

Proof: ABCD is a parallelogram, and is a right angle. Since ABCD is a parallelogram and has one right 1. (Given) angle, then it has four right angles. So by the 2. (SSS) definition of a rectangle, ABCD is a rectangle. 3. (CPCTC) ANSWER: 4. (Def. of ) 5. WXYZ is a parallelogram. (If both pairs of opp. sides are , then quad. is .) 6. are supplementary. (Cons. are suppl.) 7. (Def of suppl.) 8. are right angles. (If ABCD is a parallelogram, and is a right angle. 2 and suppl., each is a rt. .) Since ABCD is a parallelogram and has one right angle, then it has four right angles. So by the 9. are right angles. (If a has 1 definition of a rectangle, ABCD is a rectangle. rt. , it has 4 rt. .) 10. WXYZ is a rectangle. (Def. of rectangle) 36. If a quadrilateral has four right angles, then it is a PROOF Write a paragraph proof of each rectangle. statement. SOLUTION: 35. If a parallelogram has one right angle, then it is a You need to walk through the proof step by step. rectangle. Look over what you are given and what you need to SOLUTION: prove. Here, you are given a quadrilateral with four You need to walk through the proof step by step. right angles. You need to prove that it is a rectangle. Look over what you are given and what you need to Let ABCD be a quadrilateral with four right angles. prove. Here, you are given a parallelogram with one Then use the properties of parallelograms and right angle. You need to prove that it is a rectangle. rectangles to walk through the proof. Let ABCD be a parallelogram with one right angle. Then use the properties of parallelograms to walk through the proof.

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a

ABCD is a parallelogram, and is a right angle. rectangle, ABCD is a rectangle. Since ABCD is a parallelogram and has one right ANSWER: angle, then it has four right angles. So by the definition of a rectangle, ABCD is a rectangle.

ANSWER:

ABCD is a quadrilateral with four right angles. ABCD is a parallelogram because both pairs of opposite angles are congruent. By definition of a

ABCD is a parallelogram, and is a right angle. rectangle, ABCD is a rectangle. Since ABCD is a parallelogram and has one right 37. CONSTRUCTION Construct a rectangle using the angle, then it has four right angles. So by the construction for congruent segments and the definition of a rectangle, ABCD is a rectangle. construction for a line perpendicular to another line through a point on the line. Justify each step of the 36. If a quadrilateral has four right angles, then it is a construction. rectangle. SOLUTION: SOLUTION: Sample answer: You need to walk through the proof step by step. Step 1: Graph a line and place points P and Q on the Look over what you are given and what you need to line. prove. Here, you are given a quadrilateral with four right angles. You need to prove that it is a rectangle. Step 2: Place the compass on point P. Using the Let ABCD be a quadrilateral with four right angles. same compass setting, draw an arc to the left and Then use the properties of parallelograms and rectangles to walk through the proof. right of P that intersects .

Step 3: Open the compass to a setting greater than the distance from P to either point of intersection. Place the compass on each point of intersection and draw arcs that intersect above .

Step 4: Draw line a through point P and the ABCD is a quadrilateral with four right angles. intersection of the arcs drawn in step 3. ABCD is a parallelogram because both pairs of

opposite angles are congruent. By definition of a rectangle, ABCD is a rectangle. Step 5: Repeat steps 2 and 3 using point Q.

ANSWER: Step 6: Draw line b through point Q and the intersection of the arcs drawn in step 5.

Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. Label the point of intersection R.

Step 2: Place the compass on point P. Using the same compass setting, draw an arc to the left and Since and , the measure of right of P that intersects . angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 3: Open the compass to a setting greater than . The same compass setting was used to the distance from P to either point of intersection. Place the compass on each point of intersection and locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and draw arcs that intersect above . congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Step 4: Draw line a through point P and the Thus, PRSQ is a rectangle. intersection of the arcs drawn in step 3.

Step 5: Repeat steps 2 and 3 using point Q. ANSWER:

Step 6: Draw line b through point Q and the Sample answer: Since and , the intersection of the arcs drawn in step 5. measure of angles P and Q is 90 degrees. Lines that are perpendicular to the same line are parallel, so Step 7: Using any compass setting, put the compass at P and draw an arc above it that intersects line a. . The same compass setting was used to Label the point of intersection R. locate points R and S, so . If one pair of opposite sides of a quadrilateral is both parallel and Step 8: Using the same compass setting, put the congruent, then the quadrilateral is a parallelogram. compass at Q and draw an arc above it that A parallelogram with right angles is a rectangle. intersects line b. Label the point of intersection S. Thus, PRSQ is a rectangle.

Step 9: Draw .

ANSWER: Sample answer: Since and , the Sample answer: He should measure the diagonals of measure of angles P and Q is 90 degrees. Lines that the end zone and both pairs of opposite sides. If the are perpendicular to the same line are parallel, so diagonals and both pairs of opposite sides are . The same compass setting was used to congruent, then the end zone is a rectangle. locate points R and S, so . If one pair of ALGEBRA Quadrilateral WXYZ is a rectangle. opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. A parallelogram with right angles is a rectangle. Thus, PRSQ is a rectangle.

38. SPORTS The end zone of a football field is 160 feet Therefore, YW = 5. wide and 30 feet long. Kyle is responsible for painting the field. He has finished the end zone. ANSWER: Explain how Kyle can confirm that the end zone is 5 the regulation size and be sure that it is also a rectangle using only a tape measure. 40. If XZ = 2c and ZY = 6, and XY = 8, find WY. SOLUTION: SOLUTION: Sample answer: If the diagonals of a parallelogram The diagonals of a rectangle are congruent to each are congruent, then it is a rectangle. He should other. So, WY = XZ = 2c. All four angles of a measure the diagonals of the end zone and both rectangle are right angles. So, is a right pairs of opposite sides. If both pairs of opposite sides 2 2 are congruent, then the quadrilateral is a triangle. By the Pythagorean Theorem, XZ = XY + 2 parallelogram. If the diagonals are congruent, then ZY . the end zone is a rectangle. ANSWER: Sample answer: He should measure the diagonals of the end zone and both pairs of opposite sides. If the diagonals and both pairs of opposite sides are congruent, then the end zone is a rectangle. Therefore, WY = 2(5) = 10.

ALGEBRA Quadrilateral WXYZ is a rectangle. ANSWER: 10

SOLUTION:

SOLUTION: ANSWER: The diagonals of a rectangle are congruent to each No; sample answer: Both pairs of opposite sides are other. So, WY = XZ = 2c. All four angles of a congruent, so the sign is a parallelogram, but no rectangle are right angles. So, is a right measure is given that can be used to prove that it is a triangle. By the Pythagorean Theorem, XZ2 = XY2 + rectangle. 2 ZY . PROOF Write a coordinate proof of each statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let Therefore, WY = 2(5) = 10. the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, ANSWER: b), and C(a, 0). You need to walk through the proof 10 step by step. Look over what you are given and what you need to prove. Here, you are given ABCD is a 41. SIGNS The sign below is in the foyer of Nyoko’s school. Based on the dimensions given, can Nyoko rectangle and you need to prove . Use the be sure that the sign is a rectangle? Explain your properties that you have learned about rectangles to reasoning. walk through the proof.

Given: ABCD is a rectangle. Prove:

SOLUTION:

rectangle. Given: ABCD is a rectangle. PROOF Write a coordinate proof of each Prove: statement. 42. The diagonals of a rectangle are congruent. SOLUTION: Begin by positioning quadrilateral ABCD on a coordinate plane. Place vertex D at the origin. Let the length of the bases be a units and the height be b units. Then the rest of the vertices are A(0, b), B(a, b), and C(a, 0). You need to walk through the proof Proof: step by step. Look over what you are given and what Use the Distance Formula to find you need to prove. Here, you are given ABCD is a and . rectangle and you need to prove . Use the have the same length, so they are properties that you have learned about rectangles to walk through the proof. congruent.

Proof: and what you need to prove. Here, you are given Use the Distance Formula to find the lengths of the ABCD and and you need to prove that diagonals. ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and The diagonals, have the same length, so Prove: ABCD is a rectangle. they are congruent.

ANSWER: Given: ABCD is a rectangle. Prove:

Proof:

AC = BD. So, Proof: Use the Distance Formula to find and . have the same length, so they are congruent.

43. If the diagonals of a parallelogram are congruent, then it is a rectangle. Because A and B are different points, SOLUTION: a ≠ 0. Then b = 0. The slope of is undefined Begin by positioning parallelogram ABCD on a and the slope of . Thus, . ∠DAB is coordinate plane. Place vertex A at the origin. Let a right angle and ABCD is a rectangle. the length of the bases be a units and the height be c units. Then the rest of the vertices are B(a, 0), C(b + ANSWER: a, c), and D(b, c). You need to walk through the Given: ABCD and proof step by step. Look over what you are given Prove: ABCD is a rectangle. and what you need to prove. Here, you are given ABCD and and you need to prove that ABCD is a rectangle. Use the properties that you have learned about rectangles to walk through the proof.

Given: ABCD and Proof: Prove: ABCD is a rectangle.

But AC = BD. So,

Proof:

AC = BD. So, Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined

and the slope of . Thus, . ∠DAB is a right angle and ABCD is a rectangle..

44. MULTIPLE REPRESENTATIONS In the problem, you will explore properties of other special parallelograms. Because A and B are different points, a. GEOMETRIC Draw three parallelograms, each a ≠ 0. Then b = 0. The slope of is undefined with all four sides congruent. Label one parallelogram ABCD, one MNOP, and one WXYZ. and the slope of . Thus, . ∠DAB is Draw the two diagonals of each parallelogram and a right angle and ABCD is a rectangle. label the intersections R. b. TABULAR ANSWER: Use a protractor to measure the appropriate angles and complete the table below. Given: ABCD and Prove: ABCD is a rectangle.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer:

Proof:

But AC = BD. So,

b. Use a protractor to measure each angle listed in the table. Because A and B are different points, a ≠ 0. Then b = 0. The slope of is undefined and the slope of . Thus, . ∠DAB is c. Sample answer: Each of the angles listed in the a right angle and ABCD is a rectangle.. table are right angles. These angles were each formed by diagonals. The diagonals of a 44. MULTIPLE REPRESENTATIONS In the parallelogram with four congruent sides are problem, you will explore properties of other special perpendicular.

parallelograms. ANSWER: a. GEOMETRIC Draw three parallelograms, each a. with all four sides congruent. Label one Sample answer: parallelogram ABCD, one MNOP, and one WXYZ. Draw the two diagonals of each parallelogram and label the intersections R. b. TABULAR Use a protractor to measure the appropriate angles and complete the table below.

c. VERBAL Make a conjecture about the diagonals of a parallelogram with four congruent sides. SOLUTION: a. Sample answer: b.

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular.

45. CHALLENGE In rectangle ABCD, Find the values of x and y.

b. Use a protractor to measure each angle listed in the table.

ANSWER: a. Sample answer: The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle with So,

The sum of the angles of a triangle is 180. So, By the Vertical Angle Theorem,

c. Sample answer: The diagonals of a parallelogram with four congruent sides are perpendicular. ANSWER: 45. CHALLENGE In rectangle ABCD, x = 6, y = −10 Find the 46. CCSS CRITIQUE Parker says that any two values of x and y. congruent acute triangles can be arranged to make a rectangle. Tamika says that only two congruent right triangles can be arranged to make a rectangle. Is either of them correct? Explain your reasoning. SOLUTION: When two congruent triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle.

SOLUTION: All four angles of a rectangle are right angles. So,

In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right angle.

So, Tamika is correct. Then the rectangles formed are ABXW, ACYW, BCYX, BDZX, CDZY, and ADZW. Therefore, 6 ANSWER: rectangles are formed by the intersecting lines. Tamika; Sample answer: When two congruent ANSWER: triangles are arranged to form a quadrilateral, two of the angles are formed by a single vertex of a triangle. 6 In order for the quadrilateral to be a rectangle, one of the angles in the congruent triangles has to be a right 48. OPEN ENDED Write the equations of four lines angle. having intersections that form the vertices of a rectangle. Verify your answer using coordinate 47. REASONING In the diagram at the right, lines n, p, geometry. q,and r are parallel and lines l and m are parallel. SOLUTION: How many rectangles are formed by the intersecting Sample answer: lines? A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis. Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 sides.

ANSWER: definition, the parallelogram is a rectangle. 6 ANSWER: Sample answer: x = 0, x = 6, y = 0, 48. OPEN ENDED Write the equations of four lines y = 4; having intersections that form the vertices of a rectangle. Verify your answer using coordinate geometry. SOLUTION: Sample answer: A rectangle has 4 right angles and each pair of opposite sides is parallel and congruent. Lines that run parallel to the x-axis will be parallel to each other and perpendicular to lines parallel to the y-axis.

Selecting y = 0 and x = 0 will form 2 sides of the rectangle. The lines y = 4 and x = 6 form the other 2 The length of is 6 − 0 or 6 units and the length of sides. is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the x = 0, x = 6, y = 0, y = 4; quadrilateral is both parallel and congruent, by Theorem 6.12, the quadrilateral is a parallelogram. Since is horizontal and is vertical, the lines are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has one right angle, it has four right angles. Therefore, by definition, the parallelogram is a rectangle.

49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not rectangles. The length of is 6 − 0 or 6 units and the length of SOLUTION: is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the Sample answer: A parallelogram is a quadrilateral in quadrilateral is both parallel and congruent, by which each pair of opposite sides are parallel. By Theorem 6.12, the quadrilateral is a parallelogram. definition of rectangle, all rectangles have both pairs of opposite sides parallel. So rectangles are always Since is horizontal and is vertical, the lines parallelograms. are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has Parallelograms with right angles are rectangles, so one right angle, it has four right angles. Therefore, by some parallelograms are rectangles, but others with definition, the parallelogram is a rectangle. non-right angles are not.

50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = 13, what values of x and y make parallelogram FGHJ a rectangle?

The length of is 6 − 0 or 6 units and the length of is 6 – 0 or 6 units. The slope of AB is 0 and the slope of DC is 0. Since a pair of sides of the A x = 3, y = 4 quadrilateral is both parallel and congruent, by B x = 4, y = 3 Theorem 6.12, the quadrilateral is a parallelogram. C x = 7, y = 8 Since is horizontal and is vertical, the lines D x = 8, y = 7 are perpendicular and the measure of the angle they form is 90. By Theorem 6.6, if a parallelogram has SOLUTION: one right angle, it has four right angles. Therefore, by The opposite sides of a parallelogram are congruent. definition, the parallelogram is a rectangle. So, FJ = GH. –3x + 5y = 11 FJ = GH 49. WRITING IN MATH Explain why all rectangles are parallelograms, but all parallelograms are not If the diagonals of a parallelogram are congruent, rectangles. then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a SOLUTION: rectangle, then will be an isosceles triangle Sample answer: A parallelogram is a quadrilateral in with That is, 3x + y = 13. which each pair of opposite sides are parallel. By definition of rectangle, all rectangles have both pairs Add the two equations to eliminate the x-term and of opposite sides parallel. So rectangles are always solve for y.

parallelograms. –3x + 5y = 11

3x + y = 13 Parallelograms with right angles are rectangles, so 6y = 24 Add to eliminate x-term. some parallelograms are rectangles, but others with y = 4 Divide each side by 6. non-right angles are not.

ANSWER: Use the value of y to find the value of x.

Sample answer: All rectangles are parallelograms 3x + 4 = 13 Substitute.

because, by definition, both pairs of opposite sides 3x = 9 Subtract 4 from each side. are parallel. Parallelograms with right angles are x = 3 Divide each side by 3. rectangles, so some parallelograms are rectangles, but others with non-right angles are not. So, x = 3, y = 4, the correct choice is A.

50. If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM = ANSWER: 13, what values of x and y make parallelogram A FGHJ a rectangle? 51. ALGEBRA A rectangular playground is surrounded by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following equations could be used to find r, the shorter side of the playground? A x = 3, y = 4 F 10r + r = 80 B x = 4, y = 3 G 4r + 10 = 80 C x = 7, y = 8 H r(r + 10) = 80 D x = 8, y = 7 J 2(r + 10) + 2r = 80 SOLUTION: SOLUTION: The opposite sides of a parallelogram are congruent. The formula to find the perimeter of a rectangle is So, FJ = GH. given by P = 2(l + w). The perimeter is given to be 8 –3x + 5y = 11 FJ = GH ft. The width is r and the length is r + 10. So, 2r + 2 (r + 10) = 80. If the diagonals of a parallelogram are congruent, Therefore, the correct choice is J. then it is a rectangle. Since FGHJ is a parallelogram, the diagonals bisect each other. So, if it is a ANSWER: rectangle, then will be an isosceles triangle J with That is, 3x + y = 13. 52. SHORT RESPONSE What is the measure of

? Add the two equations to eliminate the x-term and solve for y. –3x + 5y = 11 3x + y = 13 6y = 24 Add to eliminate x-term. y = 4 Divide each side by 6.

Use the value of y to find the value of x. SOLUTION: 3x + 4 = 13 Substitute. The diagonals of a rectangle are congruent and

3x = 9 Subtract 4 from each side. bisect each other. So, is an isosceles triangle

x = 3 Divide each side by 3. with

So, So, x = 3, y = 4, the correct choice is A. By the Exterior Angle Theorem, ANSWER: A ANSWER: 51. ALGEBRA A rectangular playground is surrounded 112 by an 80-foot fence. One side of the playground is 10 feet longer than the other. Which of the following 53. SAT/ACT If p is odd, which of the following must equations could be used to find r, the shorter side of also be odd? the playground? A 2p F 10r + r = 80 B 2p + 2

52. SHORT RESPONSE What is the measure of Choice C ? If p is odd , is not a whole number. Choice D 2p – 2 This can be rewritten as 2(p - 1) which is also a multiple of 2 so it is even. Choice E p + 2 Adding 2 to an odd number results in an odd number. This is odd. SOLUTION: The correct choice is E. The diagonals of a rectangle are congruent and bisect each other. So, is an isosceles triangle ANSWER: with E So, By the Exterior Angle Theorem, ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

ANSWER: 112

53. SAT/ACT If p is odd, which of the following must 54. also be odd? A 2p SOLUTION: B 2p + 2 The opposite sides of a parallelogram are congruent.

54. SOLUTION: ANSWER: The opposite sides of a parallelogram are congruent. x = 8, y = 22

In a parallelogram, consecutive interior angles are supplementary. 56.

ANSWER: x = 2, y = 7 55. 57. COORDINATE GEOMETRY Find the SOLUTION: coordinates of the intersection of the diagonals of The opposite sides of a parallelogram are congruent. parallelogram ABCD with vertices A(1, 3), B(6, 2), C (4, –2), and D(–1, –1). SOLUTION: The opposite sides of a parallelogram are parallel to The diagonals of a parallelogram bisect each other, each other. So, the angles with the measures (3y + so the intersection is the midpoint of either diagonal. 3)° and (4y – 19)° are alternate interior angles and Find the midpoint of or . hence they are equal.

Therefore, the coordinates of the intersection of the diagonals are (2.5, 0.5).

6-4 RectanglesANSWER: ANSWER: x = 8, y = 22 (2.5, 0.5)

Refer to the figure.

56. SOLUTION: The diagonals of a parallelogram bisect each other. So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6.

Solve the first equation to find the value of y. 2y = 14 58. If , name two congruent angles. y = 7 SOLUTION: Use the value of y in the second equation and solve for x. Since is an isosceles triangle. So, 7 + 3 = 2x + 6 4 = 2x 2 = x ANSWER:

ANSWER: x = 2, y = 7 59. If , name two congruent segments. 57. COORDINATE GEOMETRY Find the SOLUTION: coordinates of the intersection of the diagonals of In a triangle with two congruent angles, the sides parallelogram ABCD with vertices A(1, 3), B(6, 2), C opposite to the congruent angles are also congruent. (4, –2), and D(–1, –1). So, . SOLUTION: The diagonals of a parallelogram bisect each other, ANSWER: so the intersection is the midpoint of either diagonal. Find the midpoint of or . 60. If , name two congruent segments. SOLUTION:

Refer to the figure. 61. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

ANSWER:

58. If , name two congruent angles. SOLUTION: Find the distance between each pair of points. 62. (4, 2), (2, –5) Since is an isosceles triangle. So, eSolutions Manual - Powered by Cognero SOLUTION: Page 22 Use the Distance Formula to find the distance ANSWER: between the two points.

59. If , name two congruent segments. SOLUTION: In a triangle with two congruent angles, the sides ANSWER: opposite to the congruent angles are also congruent.

So, .

ANSWER: 63. (0, 6), (–1, –4)

SOLUTION: Use the Distance Formula to find the distance

60. If , name two congruent segments. between the two points. SOLUTION: In a triangle with two congruent angles, the sides opposite to the congruent angles are also congruent. So, .

ANSWER: ANSWER:

64. (–4, 3), (3, –4) SOLUTION: Use the Distance Formula to find the distance between the two points.

ANSWER:

61. If , name two congruent angles. SOLUTION: Since is an isosceles triangle. So,

6-4 RectanglesANSWER:

Find the distance between each pair of points. 62. (4, 2), (2, –5) SOLUTION: Use the Distance Formula to find the distance between the two points.

ANSWER:

63. (0, 6), (–1, –4) SOLUTION: Use the Distance Formula to find the distance between the two points.

ANSWER:

64. (–4, 3), (3, –4) SOLUTION: Use the Distance Formula to find the distance between the two points.

ANSWER: