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Home , Base (geometry), Diagonal, Isosceles trapezoid, Parallelogram, Trapezoid

Chapter 6 Study Guide and Review

State whether each sentence is true or false. If 4. The of a is one of the sides. false, replace the underlined word or phrase to SOLUTION: make a true sentence. 1. No in an are congruent. One of the parallel sides of a trapezoid is its base. The statement is true. SOLUTION: By definition, an isosceles trapezoid is a trapezoid in ANSWER: which the legs are congruent, both pairs of base true angles are congruent, and the are 5. The diagonals of a are . congruent. SOLUTION: Thus, the statement "No angles in an isosceles A rhombus has perpendicular diagonals. The trapezoid are congruent." is false. It should be "both statement is true. pairs of base angles". ANSWER: ANSWER: true false, both pairs of base angles 6. The of a trapezoid is the segment that connects the of the legs. 2. If a is a , then the diagonals are congruent. SOLUTION: SOLUTION: A diagonal is a segment that connects any two nonconsecutive vertices. The midsegment of a A rectangle is a parallelogram with four right angles, trapezoid is the segment that connects the of opposite sides parallel, opposite sides congruent, the legs of the trapezoid. opposite angles congruent, consecutive angles are

supplementary, and the diagonals bisect each other. Thus, the statement "The diagonal of a trapezoid is The statement is true. the segment that connects the midpoints of the legs." ANSWER: is false. It should be the "midsegment". true ANSWER: 3. A midsegment of a trapezoid is a segment that false, midsegment connects any two nonconsecutive vertices. 7. A rectangle is not always a parallelogram. SOLUTION: SOLUTION: The midsegment of a trapezoid is the segment that By definition, a rectangle is a parallelogram with four connects the midpoints of the legs of the trapezoid. A right angles. diagonal is a segment that connects any two

nonconsecutive vertices. Thus, the statement "A rectangle is not always a

parallelogram." is false. It should be "always". Thus, the statement "A midsegment of a trapezoid is a segment that connects any two nonconsecutive ANSWER: vertices." is false. It should be "diagonal". false, is always ANSWER: false, diagonal eSolutions Manual - Powered by Cognero Page 1 Chapter 6 Study Guide and Review

8. A with only one set of parallel sides is a Find the sum of the measures of the interior parallelogram. angles of each convex . 11. SOLUTION: A parallelogram is a quadrilateral with both pairs of SOLUTION: opposite sides parallel. A trapezoid is a quadrilateral A decagon has ten sides. Use the Polygon Interior with exactly one pair of parallel sides. Angles Sum Theorem to find the sum of its interior measures. Thus, the statement "A quadrilateral with only one set of parallel sides is a parallelogram." is false. It should Substitute n = 10 in . be "trapezoid".

ANSWER: false, trapezoid

9. Explain how to prove that a given parallelogram is a ANSWER: . 1440 SOLUTION: Show that the parallelogram is both a rectangle and 12. 15-gon a rhombus. SOLUTION: ANSWER: A 15-gon has fifteen sides. Use the Polygon Interior Show that the parallelogram is both a rectangle and Angles Sum Theorem to find the sum of its interior a rhombus. angle measures.

10. Explain how to determine the number of sides in a Substitute n = 15 in . given the measure of one of its interior angles.

SOLUTION: Multiply the given angle measure by n. Set this expression equal to (n – 2) · 180 and solve for n. ANSWER: ANSWER: 2340 Multiply the given angle measure by n. Set this expression equal to (n – 2) · 180 and solve for n.

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13. SNOWFLAKES The snowflake decoration shown 15. ≈ 166.15 is a regular . Find the sum of the measures of SOLUTION: the interior angles of the hexagon. Let n = the number of sides in the polygon. Since all angles of a regular polygon are congruent, the sum of the interior angle measures is about 166.15n. By the Polygon Interior Angles Sum Theorem, the sum of the interior angle measures can also be expressed as .

SOLUTION: A hexagon has six sides. Use the Polygon Interior ANSWER: Angles Sum Theorem to find the sum of its interior angle measures. 26 Use ABCD to find each measure. Substitute n = 6 in .

ANSWER: 16. 720 SOLUTION: We know that consecutive angles in a parallelogram The measure of an interior angle of a regular are supplementary. polygon is given. Find the number of sides in So, the polygon. Substitute. 14. 135

SOLUTION: Let n = the number of sides in the polygon. Since all ANSWER: angles of a regular polygon are congruent, the sum of 65° the interior angle measures is 135n. By the Polygon Interior Angles Sum Theorem, the sum of the interior angle measures can also be expressed as . 17. AD SOLUTION: We know that opposite sides of a parallelogram are congruent. So,

ANSWER: ANSWER: 8 18

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18. AB ALGEBRA Find the value of each variable in each parallelogram. SOLUTION: We know that opposite sides of a parallelogram are congruent. So, 20.

ANSWER: SOLUTION: 12 Since the opposite sides of a parallelogram are congruent, 2x + 9 = 4x – 5. 19. SOLUTION: Solve for x.

We know that opposite angles of a parallelogram are congruent. So,

ANSWER: 115° Since the sum of the measures of the angles in a is 180, write and solve an equation.

So, x = 7 and y = 8.

ANSWER: x = 7, y = 8

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22. DESIGN What type of information is needed to determine whether the that make up the stained glass window below are ?

21.

SOLUTION:

Since the opposite sides are congruent, 3y + 13 = 2y + 19. SOLUTION: Solve for y. If both pairs of opposite sides are the same or 3y + 13 = 2y + 19 if one pair of opposite sides is congruent and parallel, y = 6 then the shapes are parallelograms. The shapes can also be parallelograms if both pairs of opposite angles Since the opposite angles are congruent, 2x + 41 = are congruent or if the diagonals bisect each other. 115. ANSWER: Solve for x. Sample answer: If both pairs of opposite sides are the 2x + 41 = 115 same length or if one pair of opposite sides is 2x = 74 congruent and parallel, then the shapes are x = 37 parallelograms. The shapes can also be ANSWER: parallelograms if both pairs of opposite angles are x = 37, y = 6 congruent or if the diagonals bisect each other. Determine whether each quadrilateral is a parallelogram. Justify your answer.

23.

SOLUTION: One pair of opposite sides are parallel and congruent. By Theorem 6.12, if one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. No other information is needed to determine that the figure is a parallelogram.

ANSWER: yes, Theorem 6.12

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2. AE = CF (Def. of segs) 3. (Opp. sides of a ) 4. BC = AD (Def. of segs) 24. 5. BC = BF + CF, AD = AE +ED (Seg. Add. Post.) SOLUTION: 6. BF + CF = AE + ED (Subst.) The diagonals of the figure bisect each other. By 7. BF + AE = AE + ED (Subst.) Theorem 6.11, if the diagonals of a quadrilateral 8. BF = ED (Subt. Prop.) bisect each other, then the quadrilateral is a 9. (Def. of segs) parallelogram. No other information is needed to 10. (Def. of ) determine that the figure is a parallelogram. 11. Quadrilateral EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it ANSWER: is a parallelogram.) yes, Theorem 6.11 ANSWER: 25. PROOF Write a two-column proof. Given:

Prove: Quadrilateral EBFD is a parallelogram. Given: Prove: Quadrilateral EBFD is a parallelogram.

Proof: SOLUTION: 1. ABCD is a parallelogram, (Given) You need to walk through the proof step by step. 2. AE = CF (Def. of segs) Look over what you are given and what you need to 3. (Opp. sides of a ) prove. Here, you are given . You 4. BC = AD (Def. of segs) need to prove that EBFD is a parallelogram. Use the 5. BC = BF + CF, AD = AE +ED (Seg. Add. Post.) properties that you have learned about parallelograms 6. BF + CF = AE + ED (Subst.) to walk through the proof. 7. BF + AE = AE + ED (Subst.) 8. BF = ED (Subt. Prop.) Given: 9. (Def. of segs) Prove: Quadrilateral EBFD is a parallelogram. 10. (Def. of ) 11. Quadrilateral EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it is a parallelogram.)

Proof: 1. ABCD is a parallelogram, (Given) eSolutions Manual - Powered by Cognero Page 6 Chapter 6 Study Guide and Review

ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

27.

SOLUTION: 26. We know that diagonals of a parallelogram bisect each other. So, . SOLUTION: We know that opposite angles of a parallelogram are Solve for x. congruent. So, 12x + 72 = 25x + 20 and 3y + 36 = 9y – 12.

Solve for x.

12x + 72 = 25x + 20 Alternate interior angles in a parallelogram are 72 = 13x + 20 congruent. 52 = 13x 4 = x Solve for y.

Solve for y. 5y = 60 y = 12 3y + 36 = 9y – 12 36 = 6y – 12 When x = 5 and y = 12, the quadrilateral is a 48 = 6y parallelogram. 8 = y ANSWER: When x = 4 and y = 8, the quadrilateral is a x = 5, y = 12 parallelogram.

ANSWER: x = 4, y = 8

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28. PARKING The lines of the parking space shown ALGEBRA Quadrilateral EFGH is a rectangle. below are parallel. How wide is the space (in inches)?

29. If , find .

SOLUTION: All four angles of a rectangle are right angles. So, SOLUTION: Since the distance between two parallel lines is the same, we can write the equation 6x + 12 = 5x + 20 Substitute. and then solve for x.

6x + 12 = 5x + 20 x + 12 = 20 ANSWER: x = 8 33

Substitute x = 8 in 5x + 20. 30. If , find .

5x + 20 = 5(8) + 20 SOLUTION: = 60 All four angles of a rectangle are right angles. So,

So, the length of the space is 60 inches. Substitute. ANSWER:

60 in.

ANSWER: 77

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31. If FK = 32 feet, find EG. 33. If EF = 4x – 6 and HG = x + 3, find EF.

SOLUTION: SOLUTION: We know that diagonals of a rectangle are congruent The opposite sides of a rectangle are parallel and and bisect each other. So, EG = FH, FK = KH, and congruent. Therefore, EF = HG. EK = KG. EF = HG Opp. sides of rectangle are FH = FK + KH Diagonals of a rectangle bisect congruent. each other. 4x – 6 = x + 3 Substitution. = FK + FK FK = KH, substitution 3x – 6 = 3 Subtract x from each side. = 32 + 32 Substitute. 3x = 9 Add 6 to each side. = 64 Add. x = 3 Divide each side by 3.

EG is the same length as FH so EG = 64 feet. Substitute x = 3 into 4x – 6 to find EF.

ANSWER: EF = 4x – 6 Original equation. 64 = 4(3) – 6 x = 3 = 12 – 6 Multiply. 32. Find = 6 Subtract. SOLUTION: All four angles of a rectangle are right angles. So, So, EF = 6.

ANSWER:

6 ANSWER: 180

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ALGEBRA ABCD is a rhombus. If EB = 9, AB 36. CE = 12 and , find each measure. SOLUTION:

The diagonals of a rhombus are perpendicular. Use AE to find CE.

Use the . 34. AE

SOLUTION: The diagonals of a rhombus are perpendicular. So, use the Pythagorean Theorem.

Since the length must be positive, AE = 7.9.

CE = AE = 7.9

ANSWER: Since the length must be positive, AE = 7.9. 7.9

ANSWER: 37. 7.9 SOLUTION: The diagonals are perpendicular to each other. So, in 35. the EAB, SOLUTION: All four sides of a rhombus are congruent. So, is an . Therefore, All four sides of a rhombus are congruent. So, is an isosceles triangle. Therefore,

ANSWER: ANSWER: 55 35

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38. LOGOS A car company uses the symbol shown for their logo.

If the inside space of the logo is a rhombus, what is the length of ?

SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, FG = FJ = 2.5 cm.

ANSWER: 2.5 cm

COORDINATE Given each set of vertices, determine whether QRST is a So, all sides are congruent. The quadrilateral is a rhombus, a rectangle, or a square. List all that rhombus. apply. Explain. 39. Q(12, 0), R(6, –6), S(0, 0), T(6, 6) Check to see whether we can say more: are consecutive sides perpendicular? SOLUTION: First, graph the quadrilateral.

Since the products of the slopes of consecutive sides are –1, the sides are perpendicular.

So, the quadrilateral is also a rectangle and a square.

ANSWER: Rectangle, rhombus, square; all sides are , consecutive are . Use the distance formula to find the length of each side of QRST. 40. Q(–2, 4), R(5, 6), S(12, 4), T(5, 2)

SOLUTION: First, graph the quadrilateral. eSolutions Manual - Powered by Cognero Page 11 Chapter 6 Study Guide and Review

Since , the diagonals are not congruent. So, QRST is not a rectangle. Since the figure is not a rectangle, it also cannot be a square.

Check whether the two diagonals are perpendicular.

Use the distance formula to find the length of each side of QRST. Undefined slope and 0 slope are perpendicular, so the diagonals are perpendicular. Therefore, it is a rhombus.

ANSWER: Rhombus; all sides are , diagonals are .

All the sides are congruent.

If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the of the diagonals.

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Find each measure. 42. 41. GH

SOLUTION: Figure WZYX is an isosceles trapezoid. So, each pair SOLUTION: of base angles is congruent, Use the Pythagorean Theorem.

The sum of the measures of the angles of a quadrilateral is 360.

Let . Since the length must be positive, GH = 19.2.

ANSWER: 19.2 So,

ANSWER: 68

43. DESIGN Renee designed the square tile as an art project.

a. Describe a way to determine if the in the design are isosceles. b. If the perimeter of the tile is 48 inches and the perimeter of the red square is 16 inches, what is the perimeter of one of the trapezoids?

SOLUTION: a. A trapezoid is isosceles if the legs are congruent. eSolutions Manual - Powered by Cognero Page 13 Chapter 6 Study Guide and Review

The legs of the trapezoid are part of the diagonals of the square tile. The diagonals of a square bisect opposite angles, so each base angle of a trapezoid measures 45°. One pair of sides is parallel and the base angles are congruent.

b. The perimeter of a square is given by 4s, where s is the side length. Solving 48 = 4s1 and 16 = 4s2, we find that the tile is 12 in. long on a side and the red square is 4 in. long on a side. Now, all that remains is to find the two other sides of the trapezoid.

A diagonal of the tile is in., and a diagonal of the red square is in. So, the length of each

nonparallel side of a trapezoid is in.

Add to find the perimeter of the trapezoid.

ANSWER: a. Sample answer: The legs of the trapezoids are part of the diagonals of the square. The diagonals of a square bisect opposite angles, so each base angle of a trapezoid measures 45°. One pair of sides is parallel and the base angles are congruent. b.

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