Determine Whether Each Quadrilateral Is a Parallelogram. Justify Your

Determine whether each quadrilateral is a 3. KITES Charmaine is building the kite shown below. parallelogram. Justify your answer. She wants to be sure that the string around her frame forms a parallelogram before she secures the material to it. How can she use the measures of the wooden portion of the frame to prove that the string forms a parallelogram? Explain your reasoning. 1. SOLUTION: From the figure, all 4 angles are congruent. Since each pair of opposite angles are congruent, the quadrilateral is a parallelogram by Theorem 6.10.

ANSWER: Yes; each pair of opposite angles are congruent.

SOLUTION: Charmaine can use Theorem 6.11 to determine if the string forms a parallelogram. If the diagonals of a 2. quadrilateral bisect each other, then the quadrilateral is a parallelogram, so if AP = CP and BP = DP, then SOLUTION: the string forms a parallelogram. We cannot get any information on the angles, so we cannot meet the conditions of Theorem 6.10. We ANSWER: cannot get any information on the sides, so we cannot AP = CP, BP = DP; Sample answer: If the diagonals meet the conditions of Theorems 6.9 or 6.12. One of of a quadrilateral bisect each other, then the the diagonals is bisected, but the other diagonal is not quadrilateral is a parallelogram, so if AP = CP and because it is split into unequal sides. So, the BP = DP, then the string forms a parallelogram. conditions of Theorem 6.11 are not met. Therefore, the figure is not a parallelogram.

ANSWER: No; none of the tests for are fulfilled.

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ALGEBRA Find x and y so that the quadrilateral with the given vertices. Determine quadrilateral is a parallelogram. whether the figure is a parallelogram. Justify your answer with the method indicated. 6. A(–2, 4), B(5, 4), C(8, –1), D(–1, –1); Slope Formula

4. SOLUTION: SOLUTION: Opposite angles of a parallelogram are congruent. So, and . Solve for x.

Solve for y.

ANSWER: Since the slope of ≠ slope of , ABCD is not a x = 11, y = 14 parallelogram.

SOLUTION: Opposite sides of a parallelogram are congruent. So, and . Solve for x.

ANSWER: No; both pairs of opposite sides must be parallel; Solve for y. because the slope of ≠ slope of , ABCD is not a parallelogram.

ANSWER: x = 4, y = 8

COORDINATE GEOMETRY Graph each eSolutions Manual - Powered by Cognero Page 2 6-3 Tests for Parallelograms

7. W(–5, 4), X(3, 4), Y(1, –3), Z(–7, –3); Midpoint Formula 8. Write a coordinate proof for the statement: If a SOLUTION: quadrilateral is a parallelogram, then its The midpoint of is or . diagonals bisect each other. The midpoint of is or . SOLUTION: Begin by positioning parallelogram ABCD on the So, the midpoint of . By the coordinate plane so A is at the origin and the figure is definition of midpoint, in the first quadrant. Let the length of each base be a . units so vertex B will have the coordinates (a, 0). Let the height of the parallelogram be c. Since D is Since the diagonals bisect each other, WXYZ is a further to the right than A, let its coordinates be (b, c) parallelogram. and C will be at (b + a, c). Once the parallelogram is positioned and labeled, use the midpoint formula to determine whether the diagonals bisect each other.

ANSWER:

Yes; the midpoint of . By the definition of midpoint, . Because the diagonals bisect each other, WXYZ is a parallelogram. eSolutions Manual - Powered by Cognero Page 3 6-3 Tests for Parallelograms

ANSWER:

11.

SOLUTION: The quadrilateral is not a parallelogram because none of the tests for parallelograms are fulfilled.

ANSWER: No; none of the tests for are fulfilled.

12.

SOLUTION: CONSTRUCT ARGUMENTS Determine The quadrilateral is not a parallelogram because none whether each quadrilateral is a parallelogram. of the tests for parallelograms are fulfilled. Justify your answer. ANSWER: No; none of the tests for are fulfilled.

SOLUTION: 13. The quadrilateral is a parallelogram because both pairs of opposite sides are congruent. SOLUTION: The quadrilateral is a parallelogram since the ANSWER: diagonals bisect each other. Yes; both pairs of opp. sides are ≅. ANSWER: Yes; the diagonals bisect each other.

10.

SOLUTION: 14. The quadrilateral is a parallelogram because one pair SOLUTION: of opposite sides are parallel and congruent. By The quadrilateral is not a parallelogram since none of Theorem 6.12, this quadrilateral is a parallelogram. the tests for parallelograms are fulfilled.

ANSWER: ANSWER: Yes; one pair of opp. sides is ‖ and ≅. No; none of the tests for are fulfilled. eSolutions Manual - Powered by Cognero Page 4 6-3 Tests for Parallelograms

15. PROOF If ACDH is a parallelogram, B is the 16. PROOF If WXYZ is a parallelogram, , midpoint of , and F is the midpoint of , write and M is the midpoint of , write a paragraph a flow proof to prove that ABFH is a parallelogram. proof to prove that ZMY is an isosceles triangle.

SOLUTION: SOLUTION: Given: WXYZ is a parallelogram, , and M is the midpoint of . Prove: ZMY is an isosceles triangle. Proof: Because WXYZ is a parallelogram, . M is the midpoint of , so . It is given that , so by SAS . By CPCTC, . So, ZMY is an isosceles triangle, by the definition of an isosceles triangle.

ANSWER: Given: WXYZ is a parallelogram, , and M is the midpoint of . Prove: ZMY is an isosceles triangle. ANSWER: Proof: Because WXYZ is a parallelogram, . M is the midpoint of , so . It is given that , so by SAS . By CPCTC, . So, ZMY is an isosceles triangle, by the definition of an isosceles triangle.

17. REPAIR Parallelogram lifts are used to elevate large vehicles for maintenance. In the diagram, ABEF and BCDE are parallelograms. Write a two-column proof to show that ACDF is also a parallelogram.

SOLUTION: Given: ABEF is a parallelogram; BCDE is a eSolutions Manual - Powered by Cognero Page 5 6-3 Tests for Parallelograms

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

18.

SOLUTION: Opposite sides of a parallelogram are congruent.

Solve for x. Proof: Statements (Reasons) 1. ABEF is a parallelogram; BCDE is a parallelogram. (Given) 2. (Def. of Solve for y. ) 3. (Trans. Prop.) 4. ACDF is a parallelogram. (If one pair of opp. sides is , then the quad. is a .) ANSWER: ANSWER: x = 2, y = 29 Given: ABEF is a parallelogram; BCDE is a parallelogram. Prove: ACDF is a parallelogram.

Proof: Statements (Reasons) 1. ABEF is a parallelogram; BCDE is a parallelogram. (Given) 2. (Def. of ) 3. (Trans. Prop.) 4. ACDF is a parallelogram. (If one pair of opp. sides is , then the quad. is a .) eSolutions Manual - Powered by Cognero Page 6 6-3 Tests for Parallelograms

19. 20.

SOLUTION: SOLUTION: Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent.

Solve for x. Alternate interior angles are congruent.

Solve for y. Use substitution.

ANSWER: x = 8, y = 9

Substitute in .

ANSWER: x = 30, y = 15.5

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

22.

SOLUTION: 21. Opposite angles of a parallelogram are congruent. SOLUTION: So, . We know that consecutive Diagonals of a parallelogram bisect each other. angles in a parallelogram are supplementary. So, and . So, Solve for y. Solve for x.

Substitute in . Substitute in

So, or 40.

ANSWER: ANSWER: x = 11, y = 7 x = 40, y = 20

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23.

SOLUTION: Opposite sides of a parallelogram are congruent. So, and . Solve for x in terms y.

Substitute in . Since both pairs of opposite sides are parallel, ABCD is a parallelogram.

Substitute in to solve for x.

So, x = 4 and y = 3.

ANSWER: x = 4, y = 3 ANSWER:

COORDINATE GEOMETRY Graph each Yes; slope of slope of . So, . quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify Slope of slope of . So, . your answer with the method indicated. Because both pairs of opposite sides are parallel, 24. A(–3, 4), B(4, 5), C(5, –1), D(–2, –2); Slope Formula ABCD is a parallelogram.

SOLUTION:

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25. J(–4, –4), K(–3, 1), L(4, 3), M(3, –3); Distance Formula

SOLUTION:

Since both pairs of opposite sides are not congruent, JKLM is not a parallelogram.

ANSWER: No; both pairs of opposite sides must be congruent. The distance between K and L is . The distance between L and M is . The distance between M and J is . The distance between J and K is . Because both pairs of opposite sides are not congruent, JKLM is not a parallelogram.

26. V(3, 5), W(1, –2), X(–6, 2), Y(–4, 7); Slope Formula eSolutions Manual - Powered by Cognero Page 10 6-3 Tests for Parallelograms

SOLUTION:

27. Q(2, –4), R(4, 3), S(–3, 6), T(–5, –1); Distance and Slope Formulas

SOLUTION:

Since the slope of slope of and the slope of slope of , VWXY is not a parallelogram.

Slope of slope of , so .

Since QR = ST, . So, QRST is a parallelogram. ANSWER: No; a pair of opposite sides must be parallel and congruent. Slope of , slope of ,

slope of , and slope of . Because the

slope of slope of and the slope of slope of , VWXY is not a parallelogram.

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Proof: slope of .

The slope of is 0. ANSWER: slope of . Yes; a pair of opposite sides must be parallel and congruent. Slope of slope of , so The slope of is 0. Therefore, . So by definition . , of a parallelogram, ABCD is a parallelogram. so . So, QRST is a parallelogram. ANSWER: Given: Prove: ABCD is a parallelogram.

Proof: 28. Write a coordinate proof for the statement: If both slope of . pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a The slope of is 0. parallelogram. slope of . SOLUTION: The slope of is 0. Given: Therefore, . So by definition Prove: ABCD is a parallelogram. of a parallelogram, ABCD is a parallelogram.

29. Write a coordinate proof for the statement: If a parallelogram has one right angle, it has four eSolutions Manual - Powered by Cognero Page 12 6-3 Tests for Parallelograms

right angles. slope of . SOLUTION: Given: ABCD is a parallelogram. The slope of is undefined. is a right angle. Therefore, . So, Prove: are right angles. are right angles.

30. PROOF Write a paragraph proof of Theorem 6.10.

SOLUTION: Given: Prove: ABCD is a parallelogram.

Proof:

slope of .

The slope of is undefined. slope of . Proof: Draw to form two triangles. The sum of the angles of one triangle is 180, so the sum of the The slope of is undefined. angles for two triangles is 360. So, Therefore, . So, . Because are right angles. . By substitution, . So, ANSWER: . Dividing each side by 2 Given: ABCD is a parallelogram. yields . So, the consecutive angles is a right angle. are supplementary and . Likewise, Prove: are right angles. . So,

these consecutive angles are supplementary and . Opposite sides are parallel, so ABCD is a parallelogram.

ANSWER: Given: Prove: ABCD is a parallelogram.

Proof:

slope of .

The slope of is undefined.

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DF by the Subtraction Property. by the Proof: Draw to form two triangles. The sum of definition of congruence, and by the the angles of one triangle is 180, so the sum of the Transitive Property. If both pairs of opposite sides of angles for two triangles is 360. So, a quadrilateral are congruent, then the quadrilateral is . Because a parallelogram, so BCDE is a parallelogram. By the . By definition of a parallelogram, . substitution, . So, . Dividing each side by 2 b. The scale of the copied object is . BE = CD. yields . So, the consecutive angles So, BE = 12. are supplementary and . Likewise, . So, CF = CD + DF. these consecutive angles are supplementary and = 12 + 8 . Opposite sides are parallel, so ABCD is a = 20 parallelogram. Therefore, . 31. PANTOGRAPH A pantograph is a device that can be used to copy an object and either enlarge or Write a proportion. Let x be the width of the copy. reduce it based on the dimensions of the pantograph.

Solve for x.

12x = 110 x ≈ 9.2

The width of the copy is about 9.2 in. a. If , write a paragraph proof to show that . ANSWER: b. The scale of the copied object is the ratio of CF to a. BE. If AB is 12 inches, DF is 8 inches, and the width Given: of the original object is 5.5 inches, what is the width Prove: of the copy? Proof: We are given that SOLUTION: . AC = CF a. by the definition of congruence. AC = AB + BC and CF = CD + DF by the Segment Addition Postulate Given: and AB + BC = CD + DF by substitution. Using Prove: substitution again, AB + BC = AB + DF, and BC = Proof: We are given that DF by the Subtraction Property. by the . AC = CF definition of congruence, and by the by the definition of congruence. AC = AB + BC and Transitive Property. If both pairs of opposite sides of CF = CD + DF by the Segment Addition Postulate a quadrilateral are congruent, then the quadrilateral is and AB + BC = CD + DF by substitution. Using a parallelogram, so BCDE is a parallelogram. By the substitution again, AB + BC = AB + DF, and BC = eSolutions Manual - Powered by Cognero Page 14 6-3 Tests for Parallelograms

definition of a parallelogram, . PROOF Write a two-column proof. b. about 9.2 in. 32. Theorem 6.11 SOLUTION: Given: Prove: ABCD is a parallelogram.

Proof: Statements (Reasons) 1. (Given) 2. (Vertical .) 3. (SAS) 4. (CPCTC) 5. ABCD is a parallelogram. (If both pairs of opp. sides are , then quad is a .)

ANSWER: Given: Prove: ABCD is a parallelogram.

Proof: Statements (Reasons) 1. (Given) 2. (Vertical .) 3. (SAS) 4. (CPCTC) 5. ABCD is a parallelogram. (If both pairs of opp. sides are , then quad is a .)

33. Theorem 6.12 eSolutions Manual - Powered by Cognero Page 15 6-3 Tests for Parallelograms

SOLUTION: 34. CONSTRUCTION Explain how you can use Given: Theorem 6.11 to construct a parallelogram. Then Prove: ABCD is a parallelogram. construct a parallelogram using your method. SOLUTION: By Theorem 6.11, if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Begin by drawing and bisecting a segment . Then draw a line that intersects the first segment through its midpoint D. Mark a point C Proof: on one side of this line and then construct a segment Statements (Reasons) congruent to on the other side of D. You 1. (Given) now have intersecting segments which bisect each 2. Draw . (Two points determine a line.) other. Connect point A to point C, point C to point B, 3. (If two lines are , then alt. int. point B to point E, and point E to point A to form .) ACBE.

4. (Refl. Prop.) 5. (SAS) 6. (CPCTC) 7. ABCD is a parallelogram. (If both pairs of opp. sides are , then the quad. is .)

ANSWER: Given: Prove: ABCD is a parallelogram.

ANSWER: By Theorem 6.11, if the diagonals of a quadrilateral Proof: bisect each other, then the quadrilateral is a Statements (Reasons) parallelogram. Begin by drawing and bisecting a 1. (Given) segment . Then draw a line that intersects the 2. Draw . (Two points determine a line.) first segment through its midpoint D. Mark a point C 3. (If two lines are , then alt. int. on one side of this line and then construct a segment .) congruent to on the other side of D. You 4. (Refl. Prop.) now have intersecting segments which bisect each 5. (SAS) other. Connect point A to point C, point C to point B, 6. (CPCTC) point B to point E, and point E to point A to form 7. ABCD is a parallelogram. (If both pairs of opp. ACBE. sides are , then the quad. is .) eSolutions Manual - Powered by Cognero Page 16 6-3 Tests for Parallelograms

36.

SOLUTION: From the x-coordinates of W and X, has a length of a units. Since X is on the x-axis it has coordinates (a, 0). Since horizontal segments are parallel, the endpoints of have the same y-coordinate, c. The distance from Z to Y is the same as WX, a units. Since Z is at (–b, c), what should be added to –b to REASONING Name the missing coordinates get a units? The x-coordinate of Y is a – b. Thus, the for each parallelogram. missing coordinates are Y(a – b, c) and X(a, 0).

ANSWER: Y(a – b, c), X(a, 0)

35.

SOLUTION: Since AB is on the x-axis and horizontal segments are parallel, position the endpoints of so that they have the same y-coordinate, c. The distance from D to C is the same as AB, also a + b units, let the x- coordinate of D be –b and of C be a. Thus, the missing coordinates are C(a, c) and D(–b, c).

ANSWER: C(a, c), D(–b, c)

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37. SERVICE While replacing a handrail, a contractor uses a carpenter’s square to confirm that the vertical supports are perpendicular to the top step and the ground, respectively. How can the contractor prove that the two handrails are parallel using the fewest measurements? Assume that the top step and the ground are both level. SOLUTION: Begin by positioning quadrilateral RSTV and ABCD on a coordinate plane. Place vertex R at the origin. Since ABCD is formed from the midpoints of each side of RSTV, let each length and height of RSTV be in multiples of 2. Since RSTV does not have any congruent sides or any vertical sides, let R be (0, 0), V(2c, 0), T(2d, 2b), and S(2a, 2f).

Given: RSTV is a quadrilateral. A, B, C, and D are midpoints of sides respectively. Prove: ABCD is a parallelogram.

SOLUTION: Because the two vertical rails are both perpendicular to the ground, he knows that they are parallel to each other. If he measures the distance between the two rails at the top of the steps and at the bottom of the steps, and they are equal, then one pair of sides of the quadrilateral formed by the handrails is both parallel and congruent, so the quadrilateral is a parallelogram. Because the quadrilateral is a parallelogram, the two handrails are parallel by definition. Proof: Place quadrilateral RSTV on the coordinate plane and ANSWER: label coordinates as shown. (Using coordinates that Sample answer: Because the two vertical rails are are multiples of 2 will make the computation easier.) both perpendicular to the ground, he knows that they By the Midpoint Formula, the coordinates of A, B, C, are parallel to each other. If he measures the distance between the two rails at the top of the steps and at and D are the bottom of the steps, and they are equal, then one pair of sides of the quadrilateral formed by the handrails is both parallel and congruent, so the quadrilateral is a parallelogram. Because the quadrilateral is a parallelogram, the two handrails are Find the slopes of . parallel by definition. Slope of Slope of 38. PROOF Write a coordinate proof to prove that the segments joining the midpoints of the sides of any quadrilateral form a parallelogram. eSolutions Manual - Powered by Cognero Page 18 6-3 Tests for Parallelograms

Find the slopes of .

Slope of Slope of

The slopes of are the same so the segments are parallel. Use the Distance Formula to find AB and DC.

Thus, AB = DC and . Therefore, ABCD is a parallelogram because if one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.

ANSWER: Thus, AB = DC and . Therefore, ABCD is Given: RSTV is a quadrilateral. a parallelogram because if one pair of opposite sides A, B, C, and D are midpoints of sides of a quadrilateral are both parallel and congruent, respectively. then the quadrilateral is a parallelogram. Prove: ABCD is a parallelogram. 39. MULTIPLE REPRESENTATIONS In this problem, you will explore the properties of rectangles. A rectangle is a quadrilateral with four right angles.

a. Geometric Draw three rectangles with varying lengths and widths. Label one rectangle ABCD, one MNOP, and one WXYZ. Draw the two diagonals for each rectangle.

Proof: b. Tabular Measure the diagonals of each rectangle, Place quadrilateral RSTV on the coordinate plane and and complete the table. label coordinates as shown. (Using coordinates that are multiples of 2 will make the computation easier.) By the Midpoint Formula, the coordinates of A, B, C,

and D are

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a. Sample answer:

c. Verbal Write a conjecture about the diagonals of a rectangle.

SOLUTION: a. Sample answer: b.

c. Sample answer: The diagonals of a rectangle are congruent.

40. CHALLENGE The diagonals of a parallelogram meet at the point (0, 1). One vertex of the b. Use a ruler to measure the length of each parallelogram is located at (2, 4), and a second vertex diagonal. is located at (3, 1). Find the locations of the remaining vertices.

SOLUTION: First, graph the given points. The midpoint of each diagonal is (0, 1).

c. Sample answer: The diagonals of a rectangle are congruent.

ANSWER: eSolutions Manual - Powered by Cognero Page 20 6-3 Tests for Parallelograms

ANSWER: (−3, 1) and (−2, −2)

41. WRITING IN MATH Compare and contrast Theorem 6.9 and Theorem 6.3.

SOLUTION: Theorem 6.9 states "If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram." Theorem 6.3 states "If a Let and be the coordinates of the quadrilateral is a parallelogram, then its opposite sides remaining vertices. Here, diagonals of a are congruent." The theorems are converses of each parallelogram meet at the point (0, 1). other since the hypothesis of one is the conclusion of the other. The hypothesis of Theorem 6.3 is “a figure So, and is a ”, and the hypothesis of 6.9 is “both pairs of opp. sides of a quadrilateral are ”. The conclusion . of Theorem 6.3 is “opp. sides are ”, and the conclusion of 6.9 is “the quadrilateral is a ”.

ANSWER: Consider . Sample answer: The theorems are converses of each other. The hypothesis of Theorem 6.3 is “a figure is a ”, and the hypothesis of 6.9 is “both pairs of opp. sides of a quadrilateral are ”. The conclusion of Theorem 6.3 is “opp. sides are ”, and the conclusion of 6.9 is “the quadrilateral is a ”.

Consider .

Therefore, the coordinates of the remaining vertices are (–3, 1) and (–2, –2).

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42. CONSTRUCT ARGUMENTS If two 43. OPEN-ENDED Position and label a parallelogram parallelograms have four congruent corresponding on the coordinate plane differently than shown in angles, are the parallelograms sometimes, always, or either Example 5, Exercise 35, or Exercise 36. never congruent? Justify. SOLUTION: SOLUTION: If two parallelograms have four congruent corresponding angles, then the parallelograms are sometimes congruent.

The two parallelograms could be congruent, but you can also make the parallelogram bigger or smaller without changing the angle measures by changing the side lengths. For example, these parallelograms have ANSWER: corresponding congruent angles but the parallelogram on the right is larger than the other.

ANSWER: 44. CHALLENGE If ABCD is a parallelogram and Sometimes; Sample answer: The two parallelograms , show that quadrilateral JBKD is a could be congruent, but you can also make the parallelogram. parallelogram bigger or smaller without changing the angle measures by changing the side lengths.

SOLUTION: Given: ABCD is a parallelogram and . Prove: Quadrilateral JBKD is a parallelogram.

Proof: Draw in segment . Because ABCD is a eSolutions Manual - Powered by Cognero Page 22 6-3 Tests for Parallelograms

parallelogram, then by Theorem 6.3, diagonals 45. WRITING IN MATH How can you prove that a bisect each other. Label their point of quadrilateral is a parallelogram? intersection P. By the definition of bisect, , SOLUTION: so AP = PC. By Segment Addition, AP = AJ + JP You can show that both pairs of opposite sides are and PC = PK + KC. So AJ + JP = PK + KC by congruent or parallel, both pairs of opposite angles Substitution. Because , AJ = KC by the are congruent, diagonals bisect each other, or one definition of congruence. Substituting yields KC + JP pair of opposite sides is both congruent and parallel. = PK + KC. By the Subtraction Property, JP = KC. So by the definition of congruence, . Thus, ANSWER: P is the midpoint of . Because bisect Sample answer: You can show that both pairs of each other and are diagonals of quadrilateral JBKD, opposite sides are congruent or parallel, both pairs of by Theorem 6.11, quadrilateral JBKD is a opposite angles are congruent, diagonals bisect each parallelogram. other, or one pair of opposite sides is both congruent and parallel. ANSWER: Given: ABCD is a parallelogram and . Prove: Quadrilateral JBKD is a parallelogram.

Proof: Draw in segment . Because ABCD is a parallelogram, then by Theorem 6.3, diagonals bisect each other. Label their point of intersection P. By the definition of bisect, , so AP = PC. By Segment Addition, AP = AJ + JP and PC = PK + KC. So AJ + JP = PK + KC by Substitution. Because , AJ = KC by the definition of congruence. Substituting yields KC + JP = PK + KC. By the Subtraction Property, JP = KC. So by the definition of congruence, . Thus, P is the midpoint of . Because bisect each other and are diagonals of quadrilateral JBKD, by Theorem 6.11, quadrilateral JBKD is a parallelogram.

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46. Amir is using software to design a geometric pattern 47. Quadrilateral ABCD has vertices A(0, 1), B(1, 4), for a quilt. He starts by making quadrilateral PQRS, C(5, 5), and D(4, 2). Kalina uses the slope formula to as shown. He wants to know if the quadrilateral he show that is parallel to . Which of the made is a parallelogram. following can she use as her next step to show that ABCD is a parallelogram?

I. Use the Slope Formula to show that is parallel to . II. Use the Distance Formula to show that is congruent to . III. Use the Distance Formula to show that Which of the following best describes quadrilateral is congruent to . PQRS? A I only A It is a parallelogram, because a pair of opposite B II only sides are congruent. C I and II only B It is a parallelogram, because a pair of opposite D III only angles are congruent. E I, II, and III C It is a parallelogram, because a pair of opposite SOLUTION: sides are both parallel and congruent. D There is not enough information to conclude that Since Kalina has shown that one pair of opposite the quadrilateral is a parallelogram. sides are parallel, using the Slope Formula, option I is correct as this would also prove that the other pair of SOLUTION: opposite sides are parallel. This meets the sufficient Choice A is not correct because it states that a pair conditions of the definition of a parallelogram. of opposite sides are congruent whereas Theorem 6.9 Similarly, option II would also be correct because it indicates that BOTH pairs of opposite sides need to would help her to prove that the same pair of opposite be congruent. sides are parallel and congruent. This meets the sufficient conditions for Theorem 6.12. However, Choice B is not correct because it states that a pair option III would not be correct as it would prove one of opposite angles are congruent whereas Theorem pair of opposite sides is congruent but not the same 6.10 indicates that BOTH pairs of opposite angles pair of opposite sides that she already proved parallel. need to be congruent. Therefore, since only options I and II are correct, the correct answer is choice C. Choice C is not correct because it states that a pair of opposite sides are parallel and congruent. Although ANSWER: there is a pair of congruent opposite sides, there is not C enough information available to prove that this same pair of opposite sides is also parallel.

Therefore, Choice D is the correct answer.

ANSWER: D

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48. In quadrilateral WXYZ, WX = 4x – 15, XY = 4x + 20, YZ = 3x + 5, and ZW = 6x – 20. For what value of x 3(14) + 14 = 56 is quadrilateral WXYZ a parallelogram? b. 6(14) – 16 = 68 SOLUTION: Set WX = YZ and solve for x. Then, do the same for c. 4(14) = 56 XY and ZW. d. m∠OPQ = m∠MNO = 56

e. Yes. Since ∠MNO ≅ ∠OPQ, by the alternate interior angle theorem, the line segments are parallel.

f. Yes, since both sets of opposite sides are parallel, NPQR is a parallelogram.

g. See students' work.

If x = 20, then both pairs of opposite sides are ANSWER: congruent and WXYZ is a parallelogram. The correct a. 56 answer is 20. b. 68 c. 56 ANSWER: d. 56 20 e. Sample answer: Yes. Since ∠MNO ≅ ∠OPQ, by 49. MULTI-STEP Use the figure to answer the the alternate interior angle theorem, the line segments questions. are parallel. f. Sample answer: Yes, since both sets of opposite sides are parallel, NPQR is a parallelogram. g. See students' work.

a. What is m∠MNO? b. What is m∠OQP? c. What is m∠MON? d. What is m∠OPQ? e. Is ? Explain. f. Is NPQR a parallelogram? Explain. g. What mathematical practice did you use to solve this problem?

SOLUTION: a. 3y + 14 + 4y + 6y – 16 = 180 13y = 182 y = 14 eSolutions Manual - Powered by Cognero Page 25 6-3 Tests for Parallelograms

50. Three of the vertices of a parallelogram are at (3, – 51. Which are ways to determine that a quadrilateral is a 4), (–1, –2), and (6, 1). Which point could be the parallelogram? Select all that apply. location of the fourth vertex? Select all that apply. A The quadrilateral has two pairs of congruent A (10, –1) angles. B (3, 7) C (2, 3) B The quadrilateral has both pairs of opposite sides D (–4, –7) congruent. E (–6, –1) C The quadrilateral has one pair of opposite sides SOLUTION: that are parallel and congruent. Because it is given that the figure is a parallelogram, and both pairs of opposite sides and angles are D The quadrilateral has diagonals that are different congruent, you can use the Slope Formula and lengths. Distance Formula to determine the possible locations of the fourth vertex. E The quadrilateral has both pairs of opposite angles congruent. Thus, the possible points are (10, –1), (2, 3), and (–4, –7). So, the correct answers are choices A, C, and SOLUTION: D. Considering what has been learned in this chapter, the only choices that can be used to determine that a ANSWER: quadrilateral is a parallelogram are: A, C, D The quadrilateral has both pairs of opposite sides congruent. The quadrilateral has one pair of opposite sides that are parallel and congruent. The quadrilateral has both pairs of opposite angles congruent.

So, the correct answers are choices B, C, and E.

ANSWER: B, C, E