)LQGHDFKPHDVXUH &225',1$7(*(20(75

Find each measure.1.

SOLUTION:

The trapezoid ABCD is an isosceles trapezoid. So,each pair of base angles is congruent. Therefore,

ANSWER:

101

2. WT, if ZX = 20 and TY = 15

SOLUTION:

The trapezoid WXYZ is an isosceles trapezoid. So, thediagonals are congruent. Therefore, WY = ZX.WT + TY = ZXWT + 15 = 20WT = 5

ANSWER:

5

COORDINATE GEOMETRY QuadrilateralABCD has vertices A(–4, –1), B(–2, 3), C(3, 3),and D(5, –1).

3. Verify that ABCD is a trapezoid.

SOLUTION:

First, graph the points on a coordinate grid and drawthe trapezoid.

Use the slope formula to find the slope of the sides ofthe trapezoid.

The slopes of exactly one pair of opposite sides areequal. So, they are parallel. Therefore, quadrilateralABCD is a trapezoid.

ANSWER:

ABCD is a trapezoid.

eSolutions Manual - Powered by Cognero Page 1

6-6 Trapezoids and Kites

4. Determine whether ABCD is an isosceles trapezoid.Explain.

SOLUTION:

Use the Distance Formula to find the lengths of thelegs of the trapezoid.

The lengths of the legs are equal. Therefore, ABCDis an isosceles trapezoid.

ANSWER:

isosceles;

5. In the figure, is the midsegment of trapezoidTWRV. Determine the value of x.

SOLUTION:

By the Trapezoid Midsegment Theorem, themidsegment of a trapezoid is parallel to each baseand its measure is one half the sum of the lengths ofthe bases.

are the bases and is themidsegment. So,

Solve for x. 16 = 14.8 + x1.2 = x

ANSWER:

1.2



SENSE-MAKING If ABCD is a kite, find eachmeasure.

6. AB

SOLUTION:

By the Pythagorean Theorem,

AB2 = 42 + 32 = 25

ANSWER:

5

7.

SOLUTION:

∠A is an obtuse angle and ∠C is an acute angle.Since a kite can only have one pair of oppositecongruent angles, and The sum of the measures of the angles of aquadrilateral is 360.

ANSWER:

70

Find each measure.8.

SOLUTION:

Trapezoid JKLM is an isosceles trapezoid, so eachpair of base angles is congruent. So,

The sum of the measures of the angles of aquadrilateral is 360. Let m∠J = m∠K = x.

So,

ANSWER:

100



9.

SOLUTION:

Trapezoid QRST is an isosceles trapezoid, so eachpair of base angles is congruent. So,

The sum of the measures of the angles of aquadrilateral is 360. Let m∠Q = m∠T = x.

So,

ANSWER:

70

10. JL, if KP = 4 and PM = 7

SOLUTION:

Trapezoid JKLM is an isosceles trapezoid, so thediagonals are congruent. Therefore, KM = JL. KM = KP + PM = 4 + 7 = 11 JL = KM = 11

ANSWER:

11

11. PW, if XZ = 18 and PY = 3

SOLUTION:

Trapezoid WXYZ is an isosceles trapezoid, so thediagonals are congruent. Therefore, YW = XZ. YP + PW = XZ. 3 + PW = 18 PW = 15

ANSWER:

15

COORDINATE GEOMETRY For eachquadrilateral with the given vertices, verify thatthe quadrilateral is a trapezoid and determine



whether the figure is an isosceles trapezoid.12. A(–2, 5), B(–3, 1), C(6, 1), D(3, 5)

SOLUTION:

First, graph the trapezoid.

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

The slopes of exactly one pair of opposite sides areequal, so they are parallel. Therefore, quadrilateralABCD is a trapezoid. Use the Distance Formula to find the lengths of thelegs of the trapezoid.

The lengths of the legs are not equal. Therefore,ABCD is not an isosceles trapezoid.

ANSWER:

ABCD is a trapezoid, but not

isosceles because and CD = 5.

13. J(–4, –6), K(6, 2), L(1, 3), M(–4, –1)

SOLUTION: First, graph the trapezoid.


The slopes of exactly one pair of opposite sides areequal, so they are parallel. Therefore, quadrilateralJKLM is a trapezoid. Use the Distance Formula to find the lengths of thelegs of the trapezoid.

The lengths of the legs are not equal. Therefore,JKLM is not an isosceles trapezoid.

ANSWER:

JKLM is a trapezoid, but not

isosceles because and JM = 5.eSolutions Manual - Powered by Cognero Page 5


14. Q(2, 5), R(–2, 1), S(–1, –6), T(9, 4)

SOLUTION:



The slopes of exactly one pair of opposite sides areequal, so they are parallel. Therefore, quadrilateralQRST is a trapezoid. Use the Distance Formula to find the lengths of thelegs of the trapezoid.

The lengths of the legs are equal. Therefore, QRST isan isosceles trapezoid.

ANSWER:

QRST is a trapezoid, and

isosceles because .

15. W(–5, –1), X(–2, 2), Y(3, 1), Z(5, –3)

SOLUTION:



The slopes of exactly one pair of opposite sides areequal, so they are parallel. Therefore, quadrilateralWXYZ is a trapezoid. Use the Distance Formula to find the lengths of thelegs of the trapezoid.

The lengths of the legs are not equal. Therefore,WXYZ is not an isosceles trapezoid.

ANSWER:

WXYZ is a trapezoid, but not

isosceles because .



For trapezoid QRTU, V and S are midpoints ofthe legs.

16. If QR = 12 and UT = 22, find VS.

SOLUTION:



ANSWER:

17

17. If QR = 4 and UT = 16, find VS.

SOLUTION:



ANSWER:

10

18. If VS = 9 and UT = 12, find QR.

SOLUTION:



ANSWER:

6

19. If TU = 26 and SV = 17, find QR.

SOLUTION:



ANSWER:

8



20. If QR = 2 and VS = 7, find UT.

SOLUTION:



ANSWER:

12

21. If RQ = 5 and VS = 11, find UT.

SOLUTION:



ANSWER:

17

22. DESIGN Juana is designing a window box. Shewants the end of the box to be a trapezoid with thedimensions shown. If she wants to put a shelf in themiddle for the plants to rest on, about how wideshould she make the shelf?

SOLUTION:

The length of the midsegment of the trapezoid isabout the width of the shelf. By the Trapezoid Midsegment Theorem, themidsegment of a trapezoid is parallel to each baseand its measure is one half the sum of the lengths ofthe bases. The length of the bases are 22 inches and12 inches. So, the length of the midsegment is:

Therefore, without knowing the thickness of thesides of the box, the width of the shelf is about 17inches.

ANSWER:

17 in.



23. MUSIC The keys of the xylophone shown form atrapezoid. If the length of the lower pitched C is 6inches long, and the higher pitched D is 1.8 incheslong, how long is the G key?

SOLUTION:

The G key is the midsegment of the trapezoid. By the Trapezoid Midsegment Theorem, themidsegment of a trapezoid is parallel to each baseand its measure is one half the sum of the lengths ofthe bases. So, the length of the G key is:

ANSWER:

3.9 in.

SENSE-MAKING If WXYZ is a kite, find eachmeasure.

24. YZ

SOLUTION:


XY2 = 82 + 52 = 89

A kite is a quadrilateral with exactly two pairs of

consecutive congruent sides. So,

Therefore,

ANSWER:

25. WP

SOLUTION:


WP2 = WX2 – XP2 = 62 – 42 = 20

ANSWER:



26.

SOLUTION:

A kite can only have one pair of opposite congruent

angles and Let m∠X = m∠Z = x.The sum of the measures of the angles of aquadrilateral is 360.

So,

ANSWER:

117

27.

SOLUTION:

A kite can only have one pair of opposite congruent

angles and Let m∠X = m∠Z = x.The sum of the measures of the angles of aquadrilateral is 360.

So,

ANSWER:

75

PROOF Write a paragraph proof for eachtheorem.

28. Theorem 6.21

SOLUTION:

Given: ABCD is an isosceles trapezoid. Prove:

Proof: Draw auxiliary segments so that

. Because and

parallel lines are everywhere equidistant, .Perpendicular lines form right angles, so

are right angles. are right triangles by definition.

Therefore, by HL. byCPCTC. Because are rightangles and all right angles are congruent,

. by CPCTC. So, by angle addition.

ANSWER:

Given: ABCD is an isosceles trapezoid. Prove:

Proof: Draw auxiliary segments so that

. Because and

parallel lines are everywhere equidistant, .Perpendicular lines form right angles, so

are right angles. are right triangles by definition.



Therefore, by HL. byCPCTC. Because are rightangles and all right angles are congruent,

. by CPCTC. So, by angle addition.

29. Theorem 6.22

SOLUTION:

Given: ABCD is a trapezoid; Prove: Trapezoid ABCD is isosceles.

Proof: By the Parallel Postulate, we can draw the

auxiliary line . , by the Corr. Thm. We are given that , so by the

Trans. Prop, . So, is isosceles

and . From the def. of a trapezoid,

. Because both pairs of opposite sides are

parallel, ABED is a parallelogram. So, . By

the Transitive Property, . Thus, ABCD isan isosceles trapezoid.

ANSWER:

Given: ABCD is a trapezoid; Prove: Trapezoid ABCD is isosceles.

Proof: By the Parallel Postulate, we can draw the

auxiliary line . , by the Corr.

Thm. We are given that , so by theTrans. Prop, . So, is isosceles

and . From the def. of a trapezoid,

. Because both pairs of opposite sides are

parallel, ABED is a parallelogram. So, . By

the Transitive Property, . Thus, ABCD isan isosceles trapezoid.

30. Theorem 6.23

SOLUTION:

Given: ABCD is a trapezoid;

.Prove: trapezoid ABCD is isosceles.

Proof: ABCD is a trapezoid with . Draw

auxiliary segments so that .Because perpendicular lines form right angles,

are right angles. Therefore, are right triangles by definition.

because two lines in a plane perpendicular

to the same line are parallel since oppositesides of a trapezoid are congruent. by HL and by CPCTC. Because

by the Reflexive Property of Congruence,

(SAS). by CPCTC, sotrapezoid ABCD is isosceles.

ANSWER:

Given: ABCD is a trapezoid;

.Prove: trapezoid ABCD is isosceles.



Proof: ABCD is a trapezoid with . Draw

auxiliary segments so that .Because perpendicular lines form right angles,

are right angles. Therefore, are right triangles by definition.

because two lines in a plane perpendicular

to the same line are parallel since oppositesides of a trapezoid are congruent. by HL and by CPCTC. Because

by the Reflexive Property of Congruence,

(SAS). by CPCTC, sotrapezoid ABCD is isosceles.

31. Theorem 6.25

SOLUTION:

Given: ABCD is a kite with

.

Prove:

Proof: We know that . So,B and D are both equidistant from A and C. If a pointis equidistant from the endpoints of a segment, then itis on the perpendicular bisector of the segment. Theline that contains B and D is the perpendicular

bisector of , because only one line exists through

two points. Thus, .

ANSWER:

Given: ABCD is a kite with

.

Prove:

Proof: We know that . So,B and D are both equidistant from A and C. If a pointis equidistant from the endpoints of a segment, then itis on the perpendicular bisector of the segment. Theline that contains B and D is the perpendicular

bisector of , because only one line exists through

two points. Thus, .



32. Theorem 6.26

SOLUTION:

Given: ABCD is a kite

Prove:

Proof: We know that by the

definition of a kite. by the ReflexiveProperty. Therefore, by SSS.

by CPCTC. If , thenABCD is a parallelogram by definition, which cannotbe true because we are given that ABCD is a kite.

Therefore, .

ANSWER:

Given: ABCD is a kite

Prove:

Proof: We know that by the

definition of a kite. by the ReflexiveProperty. Therefore, by SSS.

by CPCTC. If , thenABCD is a parallelogram by definition, which cannotbe true because we are given that ABCD is a kite.

Therefore, .

33. PROOF Write a coordinate proof for Theorem 6.24.

SOLUTION:

Begin by positioning trapezoid ABCD on a coordinate

plane. Place vertex D at the origin with the longerbase along the x-axis. Let the distance from D to Abe a units, the distance from A to B b units, and thedistance from B to C c units. Let the length of thebases be a units and the height be c units. Then therest of the vertices are A(a, d), B(a + b, d), and C(a+ b + c, 0). Given: ABCD is a trapezoid with median .

Prove:

Proof: By the definition of the median of a trapezoid,

E is the midpoint of and F is the midpoint of .

Midpoint E is .

Midpoint F is

.

The slope of , the slope of , and the

slope of . Thus, .



Thus, .

ANSWER:

Given: ABCD is a trapezoid with median .

Prove:

Proof: By the definition of the median of a trapezoid,

E is the midpoint of and F is the midpoint of .

Midpoint E is .

Midpoint F is

.


slope of . Thus, .

Thus, .

34. COORDINATE GEOMETRY Refer toquadrilateral ABCD.

a. Determine whether the figure is a trapezoid. If so,is it isosceles? Explain.b. Is the midsegment contained in the line withequation y = – x + 1? Justify your answer.c. Find the length of the midsegment.

SOLUTION:

a. Use the slope formula to find the slope of the sidesof the quadrilateral.

The slopes of exactly one pair of opposite sides areequal, so they are parallel. Therefore, quadrilateralABCD is a trapezoid.



Use the Distance Formula to find the lengths of thelegs of the trapezoid.

The lengths of the legs are not equal. Therefore,ABCD is not an isosceles trapezoid. b. By the Trapezoid Midsegment Theorem, themidsegment of a trapezoid is parallel to each baseand its measure is one half the sum of the lengths ofthe bases. Here, the slope of the bases of the

trapezoid is But the slope of the line with the

equation y = –x + 1 is –1. So, they are not parallel. c. Use the Distance formula to find the lengths of thebases.

The length of the midpoint is one half the sum of thelengths of the bases. So, the length is

ANSWER:

a. ABCD is a trapezoid, but not

isosceles, because and CD = 4.b. No; it is not to the bases which have slopes of

, while y = –x + 1 has a slope of –1.

c. 7.5 units

ALGEBRA ABCD is a trapezoid.

35. If AC = 3x – 7 and BD = 2x + 8, find the value of xso that ABCD is isosceles.

SOLUTION:

Trapezoid ABCD will be an isosceles trapezoid if thediagonals are congruent. AC = BD3x – 7 = 2x + 8 x = 15 When x = 15, ABCD is an isosceles trapezoid.

ANSWER:

15

36. If , findthe value of x so that ABCD is isosceles.

SOLUTION:

Trapezoid ABCD will be an isosceles trapezoid ifeach pair of base angles is congruent. m∠ABC = m∠ABC 4x + 11 = 2x + 33 2x = 22 x = 11 When x = 11, ABCD is an isosceles trapezoid.

ANSWER:

11



SPORTS The end of the batting cage shown is anisosceles trapezoid. If PT = 12 feet, ST = 28 feet, and

, find each measure.

37. TR

SOLUTION:

Since trapezoid PQRS is an isosceles trapezoid, thediagonals are congruent. By SSS Postulate,

By CPCTC, . So, is an isosceles triangle, and TR = ST = 28 ft.

ANSWER:

28 ft

38. SQ

SOLUTION:

Since trapezoid PQRS is an isosceles trapezoid, thediagonals are congruent. By SSS Postulate,

So, by CPCTC.Therefore, is an isosceles triangle and TQ =TP = 12 ft. SQ = ST + TQ = 28 + 12 = 40 Therefore, SQ = 40 ft.

ANSWER:

40 ft

39.

SOLUTION:

Since trapezoid ABCD is an isosceles trapezoid, bothpairs of base angles are congruent. So,

Letm∠QRS = m∠PSR = x. The sum of the measures of the angles of aquadrilateral is 360.

So,

ANSWER:

70

40.

SOLUTION:

Since trapezoid ABCD is an isosceles trapezoid, eachpair of base angles is congruent. So,

ANSWER:

110



ALGEBRA For trapezoid QRST, M and P aremidpoints of the legs.

41. If QR = 16, PM = 12, and TS = 4x, find x.

SOLUTION:



Solve for x.

ANSWER:

2

42. If TS = 2x, PM = 20, and QR = 6x, find x.

SOLUTION:



Solve for x.

ANSWER:

5

43. If PM = 2x, QR = 3x, and TS = 10, find PM.

SOLUTION:



Solve for x.

ANSWER:

20



44. If TS = 2x + 2, QR = 5x + 3, and PM = 13, find TS.

SOLUTION:



Solve for x.

Substitute.TS = 2(3) + 2 = 8

ANSWER:

8

SHOPPING The side of the shopping bagshown is an isosceles trapezoid. If EC = 9inches, DB = 19 inches,

, find eachmeasure.

45. AE

SOLUTION:

Trapezoid PQRS is an isosceles trapezoid. So, thediagonals are congruent. AC = BDAE + EC = BDAE = 19 – 9 = 10 inches.

ANSWER:

10 in.

46. AC

SOLUTION:

Trapezoid PQRS is an isosceles trapezoid. So, thediagonals are congruent. AC = BD = 19 inches

ANSWER:

19 in.



47.

SOLUTION:

Trapezoid ABCD is an isosceles trapezoid. So, eachpair of base angles is congruent.

Let m∠BCD = m∠ADC = x. The sum of the measures of the angles of aquadrilateral is 360.

So,

ANSWER:

105

48.

SOLUTION:

By the Alternate Interior Angle Theorem, m∠EDC =m∠ABE = 40.

ANSWER:

40

ALGEBRA WXYZ is a kite.

49. If and find .

SOLUTION:

∠WZY is an acute angle and ∠WXY is an obtuseangle. A kite can only have one pair of oppositecongruent angles and .So, m∠ZYX = m∠ZWX= 10x. The sum of the measures of the angles of aquadrilateral is 360.

Therefore, m∠ZYX = 10(10) = 100.

ANSWER:

100



50. If and find .

SOLUTION:

∠WZY is an acute angle and ∠WXY is an obtuseangle. A kite can only have one pair of oppositecongruent angles and .So, m∠ZYX = m∠WXY= 13x + 14. The sum of the measures of the angles of aquadrilateral is 360.

Therefore, m∠ZYX = 13(7) + 14 = 105.

ANSWER:

105

CONSTRUCT ARGUMENTS Write a two-column proof.

51. Given: ABCD is an isosceles trapezoid.Prove:

SOLUTION:

Given: ABCD is an isosceles trapezoid.Prove:

Proof:Statements(Reasons)1. ABCD is an isosceles trapezoid. (Given)

2. (Def. of isos. trap.)

3. (Refl. Prop.)

4. (Diags. of isos. trap. are .)5. (SSS)6. (CPCTC)

ANSWER:

Given: ABCD is an isosceles trapezoid.Prove:

Proof:Statements(Reasons)1. ABCD is an isosceles trapezoid. (Given)

2. (Def. of isos. trap.)

3. (Refl. Prop.)

4. (Diags. of isos. trap. are .)5. (SSS)6. (CPCTC)

52. Given:

Prove: WXYV is an isosceles trapezoid.

SOLUTION:

Given: Prove: WXYV is an isosceles trapezoid.



Proof:Statements(Reasons)

1. . (Given)

2. (Mult. Prop.)

3. WX = VY (Def. of midpt.)

4. (Def. of segs.)5. (Given)

6. (If corr. , lines are .)7. WXYV is an isosceles trapezoid. (Def. of isos.trap.)

ANSWER:

Given: Prove: WXYV is an isosceles trapezoid.

Proof:Statements(Reasons)

1. . (Given)

2. (Mult. Prop.)

3. WX = VY (Def. of midpt.)

4. (Def. of segs.)5. (Given)

6. (If corr. , lines are .)7. WXYV is an isosceles trapezoid. (Def. of isos.trap.)

Determine whether each statement is always,sometimes, or never true. Explain.

53. The opposite angles of a trapezoid are supplementary.

SOLUTION:

The opposite angles of an isosceles trapezoid aresupplementary, but if a trapezoid is not isosceles, theopposite angles are not supplementary.

So, the statement is sometimes true.

ANSWER:

Sometimes; opp are supplementary in anisosceles trapezoid.

54. One pair of opposite sides are parallel in a kite.

SOLUTION:

In a kite, exactly two pairs of adjacent sides arecongruent.

So, the statement is never true.

ANSWER:

Never; exactly two pairs of adjacent sides arecongruent.



55. A square is a rhombus.

SOLUTION:

By definition, a square is a quadrilateral with 4 rightangles and 4 congruent sides. Since by definition, arhombus is a quadrilateral with 4 congruent sides, asquare is always a rhombus but a rhombus is not asquare. So, the statement is always true.

ANSWER:

Always; by def., a square is a quadrilateral with 4 rt. and 4 sides. Because by def., a rhombus is a

quadrilateral with 4 sides, a square is always arhombus.

56. A rectangle is a square.

SOLUTION:

If the rectangle has 4 congruent sides, then it is asquare. Otherwise, it is not a square.

So, the statement is sometimes true.

ANSWER:

Sometimes; if the rectangle has 4 sides, then it is asquare. Otherwise, it is not a square.

57. A parallelogram is a rectangle.

SOLUTION:

A rectangle is a parallelogram, but a parallelogram isa rectangle only if the parallelogram has 4 rightangles or congruent diagonals. So, the statement issometimes true.

ANSWER:

Sometimes; only if the parallelogram has 4 rt. and/or congruent diagonals, is it a rectangle.

58. KITES Refer to the kite. Using the properties ofkites, write a two-column proof to show that is congruent to .

SOLUTION:

Given: kite MNPQProve:

Proof:Statements (Reasons)1. MNPQ is a kite. (Given)

2. (Def. of a kite)

3. (Refl. Prop.)4. (SSS)5. (CPCTC)

6. (Refl. Prop.)7. (SAS)

ANSWER:

Sample answer:Given: kite MNPQProve:



Proof:Statements (Reasons)1. MNPQ is a kite. (Given)

2. (Def. of a kite)

3. (Refl. Prop.)4. (SSS)5. (CPCTC)

6. (Refl. Prop.)7. (SAS)

59. VENN DIAGRAM Create a Venn diagram thatincorporates all quadrilaterals, including trapezoids,isosceles trapezoids, kites, and quadrilaterals thatcannot be classified as anything other thanquadrilaterals.

SOLUTION:

To create a Venn diagram that organizes all of thequadrilaterals first review the properties of each:parallelogram, rectangle, square, rhombus,trapezoid, isosceles trapezoid, and kites. In a Venndiagram, each type of quadrilateral is represented byan oval. Ovals that overlap show a quadrilateral thatcan be classified as different quadrilaterals. Parallelograms have opposite sides that are paralleland congruent. Rectangles, squares, and rhombialso have these characteristics. So, draw a largecircle to represent parallelograms. Then look at therelationship between rectangles, squares, andrhombi. A square is a rectangle but a rectangle is notnecessarily a square. A square is a rhombus butrhombi are not necessarily squares. Twooverlapping ovals represent rectangles and rhombiwith squares in the overlapping section. Neither trapezoids nor kites are parallelograms. Akite cannot be a trapezoid and a trapezoid cannot

be a kite. So draw separate ovals to represent thesequadrilaterals. Lastly, an isosceles trapezoid is atrapezoid so draw a smaller circle inside thetrapezoid oval to represent these quadrilaterals.

ANSWER:

COORDINATE GEOMETRY Determinewhether each figure is a trapezoid, aparallelogram, a square, a rhombus, or aquadrilateral given the coordinates of thevertices. Choose the most specific term.Explain.

60. A(–1, 4), B(2, 6), C(3, 3), D(0, 1)



SOLUTION: First, graph the figure.


The slopes of each pair of opposite sides are equal,so the two pairs of opposite sides are parallel.Therefore, figure ABCD is a parallelogram. None of the adjacent sides have slopes whoseproduct is –1. So, the angles are not right angles. Use the Distance Formula to find the lengths of thesides of the parallelogram.

No consecutive sides are congruent. Therefore,figure ABCD is a parallelogram.

ANSWER:

Parallelogram; opp. sides , no rt. , noconsecutive sides .

61. W(–3, 4), X(3, 4), Y(5, 3), Z(–5, 1)

SOLUTION: First, graph the figure.

Use the slope formula to find the slope of the sides ofthe figure.

There are no parallel sides. Therefore, WXYZ can beclassified only as a quadrilateral.

ANSWER:

quadrilateral; no parallel sides

62. MULTIPLE REPRESENTATIONS In thisproblem, you will explore proportions in kites.



a. Geometric Draw a segment. Construct anoncongruent segment that perpendicularly bisectsthe first segment. Connect the endpoints of thesegments to form a quadrilateral ABCD. Repeat theprocess two times. Name the additional quadrilateralsPQRS and WXYZ. b. Tabular Copy and complete the table below.

c. Verbal Make a conjecture about a quadrilateral inwhich the diagonals are perpendicular, exactly onediagonal is bisected, and the diagonals are notcongruent.

SOLUTION:

a. Using a compass and straightedge to construct thekites ensures accuracy in that the diagonals areperpendicular and one diagonal bisects the otherdiagonal.

b. Use a ruler to measure each segment listed in thetable. Use a centimeter ruler and measure to thenearest tenth.

c. Look for the pattern in the measurements taken.Each kite has 2 pairs of congruent consecutive sides.If the diagonals of a quadrilateral are perpendicular,exactly one is bisected, and the diagonals are notcongruent, then the quadrilateral is a kite.

ANSWER:

a. Sample answer:

b.

c. If the diagonals of a quadrilateral areperpendicular, exactly one is bisected, and thediagonals are not congruent, then the quadrilateral isa kite.

PROOF Write a coordinate proof of eachstatement.



63. The diagonals of an isosceles trapezoid arecongruent.

SOLUTION:

Begin by positioning trapezoid ABCD on a coordinateplane. Place vertex A at the origin. Let the length ofthe longer base be a units, the length of the shorterbase be b units, and the height be c units. Then, therest of the vertices are B(a, 0), C(a - b, c), and D(b,c). Given: isosceles trapezoid ABCD with

Prove:

Proof:

ANSWER:

Given: isosceles trapezoid ABCD with

Prove:

Proof:

64. The median of an isosceles trapezoid is parallel to the

bases.

SOLUTION:

Begin by positioning isosceles trapezoid ABCD on acoordinate plane. Place vertex D at the origin. Letthe length of the shorter base be a units, the longerlength be a + 2b and the height be c units. Then, therest of the vertices are C(a, 0), B(a + b, c), and A(-b,c). Given: ABCD is an isosceles trapezoid with median

.

Prove:

Proof:

The midpoint of is X. The coordinates are

.

The midpoint of is Y .

Use the slope formula to find the slope of the basesand the median.

Thus, .

ANSWER:

Given: ABCD is an isosceles trapezoid with median

.

Prove:



Proof:

The midpoint of is X. The coordinates are

.

The midpoint of is Y .


slope of . Thus, .

65. ERROR ANALYSIS Bedagi and Belinda are tryingto determine in kite ABCD shown. Is either ofthem correct? Explain.

SOLUTION:

∠C is an acute angle and ∠A is an obtuse angle.Since a kite can only have one pair of oppositecongruent angles and

The sum of the measures of the angles of aquadrilateral is 360.

Therefore, Belinda is correct.

ANSWER: Belinda; m∠D = m∠B. So, m∠A + m∠B + m∠C +m∠D = 360 or m∠A + 100 + 45 + 100 = 360. So,m∠A = 115.



66. CHALLENGE If the parallel sides of a trapezoidare contained by the lines y = x + 4 and y = x – 8,what equation represents the line contained by themidsegment?

SOLUTION:

First, graph both lines.

By the Trapezoid Midsegment Theorem, themidsegment of a trapezoid is parallel to each baseand its measure is one half the sum of the lengths ofthe bases. So, the slope of the line containing themidsegment is 1. Since the midsegment is equidistantfrom both bases, the y-intercept of the linecontaining the midsegment will be the average of the

y-intercepts of the bases, Therefore,

the equation is y = x – 2.

ANSWER:

y = x – 2

67. CONSTRUCT ARGUMENTS Is it sometimes,always, or never true that a square is also a kite?Explain.

SOLUTION:

A square has all 4 sides congruent, while a kite doesnot have any opposite sides congruent. A kite hasexactly one pair of opposite angles congruent. Asquare has 4 right angles, so they are all congruent.

Therefore, the statement is never true.

ANSWER:

Never; a square has all 4 sides , while a kite doesnot have any opposite sides congruent.



68. OPEN-ENDED Sketch two noncongruenttrapezoids ABCD and FGHJ in which

.

SOLUTION:

Congruent trapezoids have corresponding sides andangles congruent. Draw the diagonals AC and BDfirst, and then connect the edges such that .Draw diagonals FH and GJ that are the same lengthas AC and BD, respectively. If the diagonals aredrawn such that the vertical angles are of differentmeasure than those in ABCD, then FG and JH willbe different than AB and DC.

ANSWER:

Sample answer:

69. WRITING IN MATH Describe the properties aquadrilateral must possess in order for thequadrilateral to be classified as a trapezoid, anisosceles trapezoid, or a kite. Compare the propertiesof all three quadrilaterals.

SOLUTION: A quadrilateral is a trapezoid if:

there is exactly one pair of sides that areparallel.

A trapezoid is an isosceles trapezoid if:

the legs are congruent.

A quadrilateral is a kite if:

there is exactly 2 pairs of congruentconsecutive sides, and

opposite sides are not congruent. A trapezoid and a kite both have four sides. In atrapezoid and isosceles trapezoid, both have exactlyone pair of parallel sides.

ANSWER:

A quadrilateral must have exactly one pair of sidesparallel to be a trapezoid. If the legs are congruent,then the trapezoid is an isosceles trapezoid. If aquadrilateral has exactly two pairs of consecutivecongruent sides with the opposite sides not congruent,the quadrilateral is a kite. A trapezoid and a kite bothhave four sides. In a trapezoid and isoscelestrapezoid, both have exactly one pair of parallel sides.



70. Quadrilateral PQRS has vertices P(−4, 1), Q(−1, 2),R(2, −2), and S(−4, −4) as shown in the graph. Whichof the following best describes the quadrilateral?

A isosceles trapezoidB kiteC parallelogramD trapezoid

SOLUTION: Looking at the graph, it can be concluded visually thatquadrilateral PQRS is neither a kite nor aparallelogram because it does not fit the definition ofeither. Therefore, quadrilateral is a trapezoid. Morespecifically, it is an isosceles trapezoid because it hasone pair of congruent sides that can be found byusing the Distance Formula. So, the correct answer ischoice A.

ANSWER: A

71. Trapezoid MNPQ has vertices M(−1, 3), N(1, 4), P(3,2), and Q(−3, −1). What is the length of themidsegment of the trapezoid? A B

C

D

SOLUTION: Graph and label points M, N, P, and Q. Calculate themidpoints of and using the MidpointFormula and sketch the midsegment formed.

Use the Distance Formula to calculate the length ofmidsegment .

So, the correct answer is choice B.

ANSWER: B

72. Quadrilateral WXYZ has vertices W(–2, 1), X(1, 2),Y(4, –2), and Z(–1, –2). Which of the following termsbest describes the quadrilateral?

A kiteB parallelogramC rhombusD trapezoid

SOLUTION: First, graph the points.



Then, find the slopes of all sides of the quadrilateralto determine if it is a parallelogram or a trapezoid.

None of the sides are parallel to each other, soquadrilateral WXYZ cannot be a parallelogram, arhombus, or a trapezoid. So, the correct answer ischoice A.

ANSWER: A

73. Quadrilateral ABCD has vertices A(–4, –1), B(1, 3),C(6, 2), and D(–4, –6). Which of the following bestdescribes the quadrilateral? A kiteB parallelogramC trapezoidD isosceles trapezoid

SOLUTION: First, graph the quadrilateral.

Looking at the quadrilateral, it can be determinedvisually that it does not meet the definitions for a kiteor a parallelogram. So, it must either be a trapezoid oran isosceles trapezoid. Again, visually it can bedetermined that no opposite sides are congruent, sothe quadrilateral is a trapezoid. The correct answer is choice C.

ANSWER: C

74. Two sides of quadrilateral FGHJ are shown.



For which coordinates of vertex J is the resultingquadrilateral not a trapezoid? A (4, −6)B (3, −3)C (5, 2)D (2, 0)E (8, 4)

SOLUTION: Graph each point and find the slopes of the sides.FGHJ is a trapezoid only if exactly one pair of sidesare parallel. Testing point J (2,0)

The slopes of and are equal. Therefore,

. Similarly, . If point (2, 0) was J,then FGHJ would have two pairs of parallel oppositesides, not just one. Therefore, (2, 0) cannot be pointJ. So, the correct answer is choice D.

ANSWER: D

75. MULTI-STEP Use the figure to answer thequestions. M is the midpoint of , N is the midpointof .

a. What is the shape of ABCD?b. What is the value of x?c. Find the measure of .d. Find the measure of .e . Which statements must be true? Select all thatapply. A B C D E f. What mathematical practice did you use to solvethis problem?

SOLUTION: a. The shape is a quadrilateral with only one pair ofparallel sides, so ABCD is a trapezoid. b. Using the properties of a trapezoid and given themidsegment:

c. Substitute.



AB = 5(5) + 10 = 35 d. Substitute. CD = 3(5) – 2 = 13 e . Given that ABCD is a trapezoid and MN is itsmidsegment, the true statements are that segmentsAB and CD and segments AB and MN are parallel,and segments AM and MD are congruent. It is notnecessarily true that segments BN and MD must becongruent, nor that segments AD and BC must becongruent. So, the correct answers are choices A, B,and E. f. See students' work.

ANSWER: a. trapezoidb. 5c. 35d. 13e . A, B, Ef. See students' work.



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