vocabulary check what is the plural of vertex · 326 chapter 6 quadrilaterals convex or concave use...

6.1 Polygons 325

1. What is the plural of vertex?

2. What do you call a polygon with 8 sides? a polygon with 15 sides?

3. Suppose you could tie a string tightly around a convex polygon. Would thelength of the string be equal to the perimeter of the polygon? What if thepolygon were concave? Explain.

Decide whether the figure is a polygon. If it is not, explain why.

4. 5. 6.

Tell whether the polygon is best described as equiangular, equilateral,regular, or none of these.

7. 8. 9.

Use the information in the diagram to find m™A.

10. 11.

RECOGNIZING POLYGONS Decide whether the figure is a polygon.

12. 13. 14.

15. 16. 17.

PRACTICE AND APPLICATIONS

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326 Chapter 6 Quadrilaterals

CONVEX OR CONCAVE Use the number of sides to tell what kind ofpolygon the shape is. Then state whether the polygon is convex or concave.

18. 19. 20.

PARACHUTES Some gym classes use parachutes that look like the polygon at the right.

21. Is the polygon a heptagon, octagon, or nonagon?

22. Polygon LMNPQRST is one name for the polygon. State two other names.

23. Name all of the diagonals that have vertex M as anendpoint. Not all of the diagonals are shown.

RECOGNIZING PROPERTIES State whether the polygon is best described asequilateral, equiangular, regular, or none of these.

24. 25. 26.

TRAFFIC SIGNS Use the number of sides of the traffic sign to tell what kindof polygon it is. Is it equilateral, equiangular, regular, or none of these?

27. 28.

29. 30.

DRAWING Draw a figure that fits the description.

31. A convex heptagon 32. A concave nonagon

33. An equilateral hexagon that is not equiangular

34. An equiangular polygon that is not equilateral

35. LOGICAL REASONING Is every triangle convex? Explain your reasoning.

36. LOGICAL REASONING Quadrilateral ABCD is regular. What is the measure of ™ABC? How do you know?

L M

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ROAD SIGNSThe shape of a sign

tells what it is for. Forexample, triangular signslike the one above are usedinternationally as warningsigns.

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FOCUS ONAPPLICATIONS

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 12–17,

48–51Example 2: Exs. 18–20,

48–51Example 3: Exs. 24–30,

48–51Example 4: Exs. 36–46

6.1 Polygons 327

ANGLE MEASURES Use the information in the diagram to find m™A.

37. 38. 39.

40. TECHNOLOGY Use geometry software to draw a quadrilateral. Measureeach interior angle and calculate the sum. What happens to the sum as you

drag the vertices of the quadrilateral?

USING ALGEBRA Use the information in the diagram to solve for x.

41. 42. 43.

44. 45. 46.

47. A decagon has ten sides and a decade has tenyears. The prefix deca- comes from Greek. It means ten. What does theprefix tri- mean? List four words that use tri- and explain what they mean.

PLANT SHAPES In Exercises 48–51, use the following information.Cross sections of seeds and fruits often resemble polygons. Next to each crosssection is the polygon it resembles. Describe each polygon. Tell what kind ofpolygon it is, whether it is concave or convex, and whether it appears to beequilateral, equiangular, regular, or none of these. � Source: The History and Folklore of

N. American Wildflowers

48. Virginia Snakeroot 49. Caraway

50. Fennel 51. Poison Hemlock

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cross section shaped like a 5 pointed star. The fruitcomes from an evergreentree whose leaflets may fold at night or when the tree is shaken.

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SOFTWARE HELPVisit our Web site

www.mcdougallittell.comto see instructions forseveral softwareapplications.

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52. MULTI-STEP PROBLEM Envelope manufacturers fold a specially-shapedpiece of paper to make an envelope, as shown below.

a. What type of polygon is formed by the outside edges of the paper before itis folded? Is the polygon convex?

b. Tell what type of polygon is formed at each step. Which of the polygonsare convex?

c. Writing Explain the reason for the V-shaped notches that are at the endsof the folds.

53. FINDING VARIABLES Find the values of x and y in the diagram at the right.Check your answer. Then copy the shape and write the measure of eachangle on your diagram.

PARALLEL LINES In the diagram, j ∞ k. Find the value of x. (Review 3.3 for 6.2)

54. 55. 56.

57. 58. 59.

COORDINATE GEOMETRY You are given the midpoints of the sides of atriangle. Find the coordinates of the vertices of the triangle. (Review 5.4)

60. L(º3, 7), M(º5, 1), N(º8, 8) 61. L(º4, º1), M(3, 6), N(º2, º8)

62. L(2, 4), M(º1, 2), N(0, 7) 63. L(º1, 3), M(6, 7), N(3, º5)

64. USING ALGEBRA Use the Distance Formula to find the lengths of thediagonals of a polygon with vertices A(0, 3), B(3, 3), C(4, 1), D(0, º1), andE(º2, 1). (Review 1.3)

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6.2 Properties of Parallelograms 333

Using Parallelograms in Real Life

FURNITURE DESIGN A drafting table is made so that the legs can be joined indifferent ways to change the slope of the drawing surface. In the arrangementbelow, the legs AC

Æand BD

Ædo not bisect each other. Is ABCD a parallelogram?

SOLUTION

No. If ABCD were a parallelogram, then by Theorem 6.5 ACÆ

would bisect BDÆ

and BDÆ

would bisect ACÆ

.

1. Write a definition of parallelogram.

Decide whether the figure is a parallelogram. If it is not, explain why not.

2. 3.

IDENTIFYING CONGRUENT PARTS Use the diagram of parallelogram JKLMat the right. Complete the statement, and give a reason for your answer.

4. JKÆ

£ � ?�� 5. MNÆ

£ � ?��

6. ™MLK £ � ?�� 7. ™JKL £ � ?��

8. JNÆ

£ � ?�� 9. KLÆ

£ � ?��

10. ™MNL £ � ?�� 11. ™MKL £ � ?��

Find the measure in parallelogram LMNQ. Explain your reasoning.

12. LM 13. LP

14. LQ 15. QP

16. m™LMN 17. m™NQL

18. m™MNQ 19. m™LMQ

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FURNITUREDESIGN

Furniture designers usegeometry, trigonometry, andother skills to create designsfor furniture.

CAREER LINKwww.mcdougallittell.com

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FINDING MEASURES Find the measure inparallelogram ABCD. Explain your reasoning.

20. DE 21. BA

22. BC 23. m™CDA

24. m™ABC 25. m™BCD

USING ALGEBRA Find the value of each variable in the parallelogram.

26. 27. 28.

29. 30. 31.

USING ALGEBRA Find the value of each variable in the parallelogram.

32. 33. 34.

35. 36. 37.

38. PROVING THEOREM 6.3 Copy and complete the proof of Theorem 6.3: If a quadrilateral is a parallelogram, then its opposite angles are congruent.

GIVEN � ABCD is a ⁄.

PROVE � ™A £ ™C,™B £ ™D

Paragraph Proof Opposite sides of a parallelogram are congruent, so ��a.�� and ��b.��. By the Reflexive Property of Congruence, ��c.��.

¤ABD £ ¤CDB because of the ��d.�� Congruence Postulate. Because ��e.�� parts of congruent triangles are congruent, ™A £ ™C.

To prove that ™B £ ™D, draw ��f.�� and use the same reasoning.

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STUDENT HELP

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39. PROVING THEOREM 6.4 Copy and complete the two-column proof ofTheorem 6.4: If a quadrilateral is a parallelogram, then its consecutive anglesare supplementary.

GIVEN � JKLM is a ⁄.

PROVE � ™J and ™K are supplementary.

You can use the same reasoning to prove any other pair of consecutive anglesin ⁄JKLM are supplementary.

DEVELOPING COORDINATE PROOF Copy and complete the coordinateproof of Theorem 6.5.

GIVEN � PORS is a ⁄.

PROVE � PRÆ

and OSÆ

bisect each other.

Plan for Proof Find the coordinates of themidpoints of the diagonals of ⁄PORS and showthat they are the same.

40. Point R is on the x-axis, and the length of ORÆ

is c units. What are the coordinates of point R?

41. The length of PSÆ

is also c units, and PSÆ

ishorizontal. What are the coordinates of point S?

42. What are the coordinates of the midpoint of PRÆ

?

43. What are the coordinates of the midpoint of OSÆ

?

44. Writing How do you know that PRÆ

and OSÆ

bisect each other?

BAKING In Exercises 45 and 46, use the following information.In a recipe for baklava, the pastry should be cut into triangles that form congruent parallelograms, as shown. Write a paragraph proof to prove the statement.

45. ™3 is supplementary to ™6.

46. ™4 is supplementary to ™5.

Statements Reasons

1. Given

2. ��?��3. Sum of measures of int.

√ of a quad. is 360°.

4. ��?��5. Distributive property

6. ��?�� prop. of equality

7. ��?��

1. ��?��2. m™J = m™L, m™K = m™M

3. m™J + m™L + m™K + m™M = ��?��

4. m™J + m™J + m™K + m™K = 360°

5. 2(��?�� + ��?��) = 360°

6. m™J + m™K = 180°

7. ™J and ™K are supplementary.

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P (a, b) S (?, ?)

HOMEWORK HELPVisit our Web site

www.mcdougallittell.comfor help with the coor-dinate proof in Exs. 40–44.

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STAIR BALUSTERS In Exercises 47–50, use the following information.In the diagram at the right, the slope of the handrail isequal to the slope of the stairs. The balusters (verticalposts) support the handrail.

47. Which angle in the red parallelogram iscongruent to ™1?

48. Which angles in the blue parallelogram aresupplementary to ™6?

49. Which postulate can be used to prove that ™1 £ ™5?

50. Writing Is the red parallelogram congruent to the blue parallelogram? Explain your reasoning.

SCISSORS LIFT Photographers can use scissors lifts for overhead shots,as shown at the left. The crossing beams of the lift form parallelogramsthat move together to raise and lower the platform. In Exercises 51–54, usethe diagram of parallelogram ABDC at the right.

51. What is m™B when m™A = 120°?

52. Suppose you decrease m™A. What happens to m™B?

53. Suppose you decrease m™A. What happens to AD?

54. Suppose you decrease m™A. What happens to the overall height of the scissors lift?

TWO-COLUMN PROOF Write a two-column proof.

55. GIVEN � ABCD and CEFD are ⁄s. 56. GIVEN � PQRS and TUVS are ⁄s.

PROVE � ABÆ

£ FEÆ

PROVE � ™1 £ ™3

57. GIVEN � WXYZ is a ⁄. 58. GIVEN � ABCD, EBGF, HJKD are ⁄s.

PROVE � ¤WMZ £ ¤YMX PROVE � ™2 £ ™3

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59. Writing In the diagram, ABCG, CDEG, andAGEF are parallelograms. Copy the diagramand add as many other angle measures as youcan. Then describe how you know the anglemeasures you added are correct.

60. MULTIPLE CHOICE In ⁄KLMN, what is thevalue of s?

¡A 5 ¡B 20 ¡C 40

¡D 52 ¡E 70

61. MULTIPLE CHOICE In ⁄ABCD, point E is the intersection of the diagonals.Which of the following is not necessarily true?

¡A AB = CD ¡B AC = BD ¡C AE = CE ¡D AD = BC ¡E DE = BE

USING ALGEBRA Suppose points A(1, 2), B (3, 6), and C (6, 4) are threevertices of a parallelogram.

62. Give the coordinates of a point that could be thefourth vertex. Sketch the parallelogram in acoordinate plane.

63. Explain how to check to make sure the figure youdrew in Exercise 62 is a parallelogram.

64. How many different parallelograms can beformed using A, B, and C as vertices? Sketcheach parallelogram and label the coordinates ofthe fourth vertex.

USING ALGEBRA Use the Distance Formula to find AB. (Review 1.3 for 6.3)

65. A(2, 1), B(6, 9) 66. A(º4, 2), B(2, º1) 67. A(º8, º4), B(º1, º3)

USING ALGEBRA Find the slope of ABÆ

. (Review 3.6 for 6.3)

68. A(2, 1), B(6, 9) 69. A(º4, 2), B(2, º1) 70. A(º8, º4), B(º1, º3)

71. PARKING CARS In a parking lot, two guidelines are painted so that theyare both perpendicular to the line along the curb. Are the guidelines parallel?Explain why or why not. (Review 3.5)

Name the shortest and longest sides of the triangle. Explain. (Review 5.5)

72. 73. 74.

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1. Is a hexagon with opposite sides parallel called a parallelogram? Explain.

Decide whether you are given enough information to determine that thequadrilateral is a parallelogram. Explain your reasoning.

2. 3. 4.

Describe how you would prove that ABCD is a parallelogram.

5. 6. 7.

8. Describe at least three ways to show that A(0, 0), B(2, 6), C(5, 7), and D(3, 1) are the vertices of a parallelogram.

LOGICAL REASONING Are you given enough information to determinewhether the quadrilateral is a parallelogram? Explain.

9. 10. 11.

12. 13. 14.

LOGICAL REASONING Describe how to prove that ABCD is aparallelogram. Use the given information.

15. ¤ABC £ ¤CDA 16. ¤AXB £ ¤CXD

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28, 31Example 3: Exs. 32, 33Example 4: Exs. 21–26,

34–36


STUDENT HELP

Concept Check ✓Skill Check ✓

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6.3 Proving Quadrilaterals are Parallelograms 343

USING ALGEBRA What value of x will make the polygon a parallelogram?

17. 18. 19.

20. VISUAL THINKING Draw a quadrilateral that has one pair of congruent sidesand one pair of parallel sides but that is not a parallelogram.

COORDINATE GEOMETRY Use the given definition or theorem to prove that ABCD is a parallelogram. Use A(º1, 6), B(3, 5), C(5, º3), and D(1, º2).

21. Theorem 6.6 22. Theorem 6.9

23. definition of a parallelogram 24. Theorem 6.10

USING COORDINATE GEOMETRY Prove that the points represent thevertices of a parallelogram. Use a different method for each exercise.

25. J(º6, 2), K(º1, 3), L(2, º3), M(º3, º4)

26. P(2, 5), Q(8, 4), R(9, º4), S(3, º3)

27. CHANGING GEARS When you change gears on a bicycle, the derailleur moves the chain to the new gear. For the derailleur at the right, AB = 1.8 cm, BC = 3.6 cm, CD = 1.8 cm, and DA = 3.6 cm. Explain why AB

Æand CD

Æare always

parallel when the derailleur moves.

28. COMPUTERS Many word processors have a feature that allows a regular letter to be changed to an oblique (slanted) letter. The diagram at the right shows some regular letters and their oblique versions. Explain how you can prove that the oblique I is a parallelogram.

29. VISUAL REASONING Explain why the following method of drawing aparallelogram works. State a theorem to support your answer.

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Draw anothersegment so themidpoints coincide.

Connect theendpoints of thesegments.

321

DERAILLEURS(named from the

French word meaning‘derail’) move the chainamong two to six sprocketsof different diameters tochange gears.

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30. CONSTRUCTION There are many ways to use a compass andstraightedge to construct a parallelogram. Describe a method that usesTheorem 6.6, Theorem 6.8, or Theorem 6.10. Then use your method toconstruct a parallelogram.

31. BIRD WATCHING You aredesigning a binocular mount that willkeep the binoculars pointed in the same direction while they are raised andlowered for different viewers. If BC

Æis

always vertical, the binoculars willalways point in the same direction. Howcan you design the mount so BC

Æis

always vertical? Justify your answer.

PROVING THEOREMS 6.7 AND 6.8 Write a proof of the theorem.

32. Prove Theorem 6.7.

GIVEN � ™R £ ™T,™S £ ™U

PROVE � RSTU is a parallelogram.

Plan for Proof Show that the sum 2(m™S) + 2(m™T) = 360°, so ™S and ™T are supplementary and SR

Æ∞ UTÆ

.

PROVING THEOREM 6.9 In Exercises 34–36, complete the coordinateproof of Theorem 6.9.

GIVEN � Diagonals MPÆ

and NQÆ

bisect each other.

PROVE � MNPQ is a parallelogram.

Plan for Proof Show that opposite sidesof MNPQ have the same slope.

Place MNPQ in the coordinate plane so thediagonals intersect at the origin and MP

Ælies on

the y-axis. Let the coordinates of M be (0, a) andthe coordinates of N be (b, c). Copy the graph atthe right.

34. What are the coordinates of P? Explain your reasoning and label thecoordinates on your graph.

35. What are the coordinates of Q? Explain your reasoning and label thecoordinates on your graph.

36. Find the slope of each side of MNPQ and show that the slopes of oppositesides are equal.

S T

R U

y

xO

M(0, a)

N(b, c)

P(?, ?)

œ(?, ?)

R

S

q

P

33. Prove Theorem 6.8.

GIVEN � ™P is supplementaryto ™Q and ™S.

PROVE � PQRS is a parallelogram.

Plan for Proof Show that oppositesides of PQRS are parallel.


www.mcdougallittell.comfor help with the coor-dinate proof in Exs. 34–36.

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A

B

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6.3 Proving Quadrilaterals are Parallelograms 345

37. MULTI-STEP PROBLEM You shoot a pool ballas shown at the right and it rolls back to whereit started. The ball bounces off each wall at thesame angle at which it hit the wall. Copy thediagram and add each angle measure as youknow it.

a. The ball hits the first wall at an angle of about 63°. So m™AEF = m™BEH ≈ 63°. Explain why m™AFE ≈ 27°.

b. Explain why m™FGD ≈ 63°.

c. What is m™GHC? m™EHB?

d. Find the measure of each interior angle ofEFGH. What kind of shape is EFGH? How do you know?

38. VISUAL THINKING PQRS is a parallelogram and QTSU is a parallelogram. Use the diagonals of the parallelograms to explain why PTRU is a parallelogram.

USING ALGEBRA Rewrite the biconditional statement as a conditionalstatement and its converse. (Review 2.2 for 6.4)

39. x2 + 2 = 2 if and only if x = 0.

40. 4x + 7 = x + 37 if and only if x = 10.

41. A quadrilateral is a parallelogram if and only if each pair of opposite sidesare parallel.

WRITING BICONDITIONAL STATEMENTS Write the pair of theorems fromLesson 5.1 as a single biconditional statement. (Review 2.2, 5.1 for 6.4)

42. Theorems 5.1 and 5.2

43. Theorems 5.3 and 5.4

44. Write an equation of the line that is perpendicular to y = º4x + 2 and passesthrough the point (1, º2). (Review 3.7)

ANGLE MEASURES Find the value of x. (Review 4.1)

45. 46. 47.

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EXTRA CHALLENGE



1. Choose the words that describe the quadrilateral at the right: concave,convex, equilateral, equiangular, and regular. (Lesson 6.1)

2. Find the value of x. Explain yourreasoning. (Lesson 6.1)

3. Write a proof. (Lesson 6.2)

GIVEN � ABCG and CDEFare parallelograms.

PROVE � ™A £ ™E

4. Describe two ways to show that A(º4, 1), B(3, 0), C(5, º7), and D(º2, º6)are the vertices of a parallelogram. (Lesson 6.3)

QUIZ 1 Self-Test for Lessons 6.1–6.3

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History of Finding Area

THENTHEN

Methods for finding areaare recorded in thisChinese manuscript.

THOUSANDS OF YEARS AGO, the Egyptians needed to find the area of the land theywere farming. The mathematical methods they used are described in a papyrusdating from about 1650 B.C.

TODAY, satellites and aerial photographs can be used to measure the areas of large or inaccessible regions.

1. Find the area of the trapezoid outlined on the aerial photograph. The formula for the area of a trapezoid appears on page 374.

APPLICATION LINKwww.mcdougallittell.com

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NOWNOW

This Egyptian papyrusincludes methods forfinding area.

Surveyors use signals fromsatellites to measure large areas.

c. 300 B.C.–A.D. 200

c.1650 B.C.1990s

1800 ft

2800 ft2800 ft

500 ft

1800 ft

6.4 Rhombuses, Rectangles, and Squares 351

1. What is another name for an equilateral quadrilateral?

2. Theorem 6.12 is a biconditional statement.Rewrite the theorem as a conditionalstatement and its converse, and tell what eachstatement means for parallelogram PQRS.

Decide whether the statement is sometimes, always, or never true.

3. A rectangle is a parallelogram. 4. A parallelogram is a rhombus.

5. A rectangle is a rhombus. 6. A square is a rectangle.

Which of the following quadrilaterals have the given property?

7. All sides are congruent.

8. All angles are congruent.

9. The diagonals are congruent.

10. Opposite angles are congruent.

11. MNPQ is a rectangle. What is the value of x?

RECTANGLE For any rectangle ABCD, decide whether the statement isalways, sometimes, or never true. Draw a sketch and explain your answer.

12. ™A £ ™B 13. ABÆ

£ BCÆ

14. ACÆ

£ BDÆ

15. ACÆ

fi BDÆ

PROPERTIES List each quadrilateral for which the statement is true.

16. It is equiangular. 17. It is equiangular and equilateral.

18. The diagonals are perpendicular. 19. Opposite sides are congruent.

20. The diagonals bisect each other. 21. The diagonals bisect opposite angles.

PROPERTIES Sketch the quadrilateral and list everything you know about it.

22. parallelogram FGHI 23. rhombus PQRS 24. square ABCD


GUIDED PRACTICE

STUDENT HELP


27–32Example 2: Exs. 27–32, 51Example 3: Exs. 33–43Example 4: Exs. 44–52Example 5: Exs. 55–60Example 6: Exs. 61, 62


STUDENT HELP

Concept Check ✓

Skill Check ✓

Vocabulary Check ✓P

S R

q

2x �M N

q P

parallelogram rectangle rhombus square

A. Parallelogram

B. Rectangle

C. Rhombus

D. Square


LOGICAL REASONING Give another name for the quadrilateral.

25. equiangular quadrilateral 26. regular quadrilateral

RHOMBUS For any rhombus ABCD, decide whether the statement isalways, sometimes, or never true. Draw a sketch and explain your answer.

27. ™A £ ™C 28. ™A £ ™B

29. ™ABD £ ™CBD 30. ABÆ

£ BCÆ

31. ACÆ

£ BDÆ

32. ADÆ

£ CDÆ

USING ALGEBRA Find the value of x.

33. ABCD is a square. 34. EFGH is a rhombus.

35. KLMN is a rectangle. 36. PQRS is a parallelogram.

37. TUWY is a rhombus. 38. CDEF is a rectangle.

COMPLETING STATEMENTS GHJK is a square with diagonals intersectingat L. Given that GH = 2 and GL = �2�, complete the statement.

39. HK = ��?��

40. m™KLJ = ��?��

41. m™HJG = ��?��

42. Perimeter of ¤HJK = ��?��

43. USING ALGEBRA WXYZ is a rectangle. The perimeter of ¤XYZ is 24.XY + YZ = 5x º 1 and XZ = 13 º x.Find WY.

xyxy

C D

F E

8x � 13

7x � 11

T U

Y Wx � 2

3x

P

S R

q10

2x

K

N

L

M(x � 40)� (2x � 10)�

E G

F

H

x �

130�

A

B

D

C

18

5x �

xyxy

G H

K J

2

L

�2

W X

Z Y


44. LOGICAL REASONING What additional informationdo you need to prove that ABCD is a square?

PROOF In Exercises 45 and 46, write any kind of proof.

45. GIVEN � MNÆ

∞ PQÆ

, ™1 ‹ ™2 46. GIVEN � RSTU is a ⁄, SUÆ

fi RTÆ

PROVE � MQÆ

is not parallel to PNÆ

. PROVE � ™STR £ ™UTR

47. BICONDITIONAL STATEMENTS Rewrite Theorem 6.13 as a conditional statement and its converse. Tell what each statement means for parallelogram JKLM.

LOGICAL REASONING Write the corollary as a conditional statement and itsconverse. Then explain why each statement is true.

48. Rhombus corollary 49. Rectangle corollary 50. Square corollary

PROVING THEOREM 6.12 Prove both conditional statements ofTheorem 6.12.

CONSTRUCTION Explain how to construct the figure using astraightedge and a compass. Use a definition or theorem from this lessonto explain why your method works.

53. a rhombus that is not a square 54. a rectangle that is not a square

G

H

J

F K

P

T

q

R

51. GIVEN � PQRT is a rhombus.

PROVE � PRÆ

bisects ™TPQ and™QRT. TQ

Æbisects ™PTR

and ™RQP.

Plan for Proof To prove that PRÆ

bisects ™TPQ and ™QRT, first prove that ¤PRQ £ ¤PRT.

52. GIVEN � FGHJ is a parallelogram.FHÆ

bisects ™JFG and™GHJ. JG

Æbisects ™FJH

and ™HGF.

PROVE � FGHJ is a rhombus.

Plan for Proof Prove ¤FHJ £¤FHG so JH

Æ£ GH

Æ. Then use the

fact that JHÆ

£ FGÆ

and GHÆ

£ FJÆ

.

R S

U T

X

M N

q P

2

1

A

BD

C

J

M

X

L

K


COORDINATE GEOMETRY It is given that PQRS is a parallelogram.Graph ⁄PQRS. Decide whether it is a rectangle, a rhombus, a square, ornone of the above. Justify your answer using theorems aboutquadrilaterals.

55. P(3, 1) 56. P(5, 2) 57. P(º1, 4) 58. P(5, 2)Q(3, º3) Q(1, 9) Q(º3, 2) Q(2, 5)R(º2, º3) R(º3, 2) R(2, º3) R(º1, 2)S(º2, 1) S(1, º5) S(4, º1) S(2, º1)

COORDINATE PROOF OF THEOREM 6.13 In Exercises 59 and 60, you willcomplete a coordinate proof of one conditional statement of Theorem 6.13.

GIVEN � KMNO is a rectangle.

PROVE � OMÆ

£ KNÆ

Because ™O is a right angle, place KMNOin the coordinate plane so O is at the origin,ONÆ

lies on the x-axis and OKÆ

lies on the y-axis. Let the coordinates of K be (0, a)and let the coordinates of N be (b, 0).

59. What are the coordinates of M? Explain your reasoning.

60. Use the Distance Formula to prove that OMÆ

£ KNÆ

.

PORTABLE TABLE The legs of the tableshown at the right are all the same length. The cross braces are all the same length andbisect each other.

61. Show that the edge of the tabletop ABÆ

isperpendicular to legs AE

Æand BF

Æ.

62. Show that ABÆ

is parallel to EFÆ

.

TECHNOLOGY In Exercises 63–65, usegeometry software.

Draw a segment ABÆ

and a point C on thesegment. Construct the midpoint D of AB

Æ. Then

hide ABÆ

and point B so only points A, D, and Care visible.

Construct two circles with centers A and C usingthe length AD

Æas the radius of each circle. Label

the points of intersection E and F. Draw AEÆ

, CEÆ

,CFÆ

, and AFÆ

.

63. What kind of shape is AECF? How do you know? What happens to the shapeas you drag A? drag C?

64. Hide the circles and point D, and draw diagonals EFÆ

and ACÆ

. Measure™EAC, ™FAC, ™AEF, and ™CEF. What happens to the measures as youdrag A? drag C?

65. Which theorem does this construction illustrate?



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STUDENT HELP

y

x

K(0, a) M(?, ?)

N(b, 0)O(0, 0)

A B

C D

E F

A

C

E

F

D


66. MULTIPLE CHOICE In rectangle ABCD, if AB = 7x º 3 and CD = 4x + 9,then x = ��? .

¡A 1 ¡B 2 ¡C 3 ¡D 4 ¡E 5

67. MULTIPLE CHOICE In parallelogram KLMN, KM = LN, m™KLM = 2xy,and m™LMN = 9x + 9. Find the value of y.

¡A 9 ¡B 5 ¡C 18

¡D 10 ¡E Cannot be determined.

68. Writing Explain why a parallelogram with one right angle is a rectangle.

COORDINATE PROOF OF THEOREM 6.13 Complete the coordinateproof of one conditional statement of Theorem 6.13.

GIVEN � ABCD is a parallelogram, ACÆ

£ DBÆ

.

PROVE � ABCD is a rectangle.

Place ABCD in the coordinate plane so DB

Ælies on the x-axis and the diagonals

intersect at the origin. Let the coordinates of B be (b, 0) and let the x-coordinate of A be a as shown.

69. Explain why OA = OB = OC = OD.

70. Write the y-coordinate of A in terms of a and b. Explain your reasoning.

71. Write the coordinates of C and D in terms of a and b. Explain your reasoning.

72. Find and compare the slopes of the sides to prove that ABCD is a rectangle.

USING THE SAS CONGRUENCE POSTULATE Decide whether enoughinformation is given to determine that ¤ABC £ ¤DEF. (Review 4.3)

73. ™A £ ™D, ABÆ

£ DEÆ

, ACÆ

£ DFÆ

74. ABÆ

£ BCÆ

, BCÆ

£ CAÆ

, ™A £ ™D

75. ™B £ ™E, ACÆ

£ DFÆ

, ABÆ

£ DEÆ

76. EFÆ

£ BCÆ

, DFÆ

£ ABÆ

, ™A £ ™E

77. ™C £ ™F, ACÆ

£ DFÆ

, BCÆ

£ EFÆ

78. ™B £ ™E, ABÆ

£ DEÆ

, BCÆ

£ EFÆ

CONCURRENCY PROPERTY FOR MEDIANS Use the information given in thediagram to fill in the blanks. (Review 5.3)

79. AP = 1, PD = ��?

80. PC = 6.6, PE = ��?

81. PB = 6, FB = ��?

82. AD = 39, PD = ��?

83. INDIRECT PROOF Write an indirect proof to show that there is noquadrilateral with four acute angles. (Review 6.1 for 6.5)

MIXED REVIEW

TestPreparation

★★ Challenge

EXTRA CHALLENGE


y

xOD(?, ?)

A(a, ?)

B(b, 0)

C(?, ?)

B

A C

DE

F

P

6.5 Trapezoids and Kites 359

1. Name the bases of trapezoid ABCD.

2. Explain why a rhombus is not a kite. Use the definition of a kite.

Decide whether the quadrilateral is a trapezoid, an isosceles trapezoid, akite, or none of these.

3. 4. 5.

6. How can you prove that trapezoid ABCD in Example 2 is isosceles?

Find the length of the midsegment.

7. 8. 9.

STUDYING A TRAPEZOID Draw a trapezoid PQRS with QRÆ

∞ PSÆ

. Identifythe segments or angles of PQRS as bases, consecutive sides, legs,diagonals, base angles, or opposite angles.

10. QRÆ

and PSÆ

11. PQÆ

and RSÆ

12. PQÆ

and QRÆ

13. QSÆ

and PRÆ

14. ™Q and ™S 15. ™S and ™P

FINDING ANGLE MEASURES Find the angle measures of JKLM.

16. 17. 18.

FINDING MIDSEGMENTS Find the length of the midsegment MNÆ

.

19. 20. 21. P M

NS

R

q

15 9

P

M N

S R

q14

16

P

MN

SR

q9

7

82�J

K L

M

78�

J

K L

M132�44�

J

K L

M


5 5

7

127

3

7

11

GUIDED PRACTICE

Vocabulary Check ✓Concept Check ✓

Skill Check ✓

A B

CD

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 16–18Example 2: Exs. 34, 37,

38, 48–50Example 3: Exs. 19–24,

35, 39Example 4: Exs. 28–30Example 5: Exs. 31–33


STUDENT HELP



22. 23. 24.

CONCENTRIC POLYGONS In the diagram, ABCDEFGHJKLM is a regulardodecagon, AB

Æ∞ PQÆ

, and X is equidistant from the vertices of thedodecagon.

25. Are you given enough information toprove that ABPQ is isosceles? Explainyour reasoning.

26. What is the measure of ™AXB?

27. What is the measure of each interiorangle of ABPQ?

USING ALGEBRA What are the lengths of the sides of the kite? Giveyour answer to the nearest hundredth.

28. 29. 30.

ANGLES OF KITES EFGH is a kite. What is m™G?

31. 32. 33.

34. ERROR ANALYSIS A student says that parallelogram ABCD is an isosceles trapezoid because AB

Æ∞ DCÆ

and ADÆ

£ BCÆ

. Explain what is wrong with this reasoning.

35. CRITICAL THINKING The midsegment of a trapezoid is 5 inches long. Whatare possible lengths of the bases?

36. COORDINATE GEOMETRY Determine whether the points A(4, 5), B(º3, 3),C(º6, º13), and D(6, º2) are the vertices of a kite. Explain your answer.

TRAPEZOIDS Determine whether the given points represent the vertices ofa trapezoid. If so, is the trapezoid isosceles? Explain your reasoning.

37. A(º2, 0), B(0, 4), C(5, 4), D(8, 0) 38. E(1, 9), F(4, 2), G(5, 2), H(8, 9)

E G

F

H

70�

100�

E

G

FH 110�

E

G

FH 120� 50�

128

5

8J L

K

M

5

4

57

E G

F

H

23

4

3A C

B

D

xyxy

x

11

8

x

4

7

x

4

7

x

9

7

xyxy

M

A

BC D

E

F

G

HJK

L

P

q

X

A B

CD

WEBS The spiderweb above is called

an orb web. Although it lookslike concentric polygons, thespider actually followed aspiral path to spin the web.

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39. LAYER CAKE The top layer of the cake has a diameter of 10 inches. The bottom layer has a diameter of 22 inches. What is the diameter of the middle layer?

40. PROVING THEOREM 6.14 Write a proof of Theorem 6.14.

GIVEN � ABCD is an isosceles trapezoid.

ABÆ

∞ DCÆ

, ADÆ

£ BCÆ

PROVE � ™D £ ™C, ™DAB £ ™B

Plan for Proof To show ™D £ ™C, first draw AEÆ

so ABCE is aparallelogram. Then show BC

Æ£ AE

Æ, so AE

Æ£ AD

Æand ™D £ ™AED.

Finally, show ™D £ ™C. To show ™DAB £ ™B, use the consecutiveinterior angles theorem and substitution.

41. PROVING THEOREM 6.16 Write a proof of one conditional statement of Theorem 6.16.

GIVEN � TQRS is an isosceles trapezoid.

QRÆ

∞ TSÆ

and QTÆ

£ RSÆ

PROVE � TRÆ

£ SQÆ

42. JUSTIFYING THEOREM 6.17 In the diagram below, BGÆ

is the midsegmentof ¤ACD and GE

Æis the midsegment of ¤ADF. Explain why the midsegment

of trapezoid ACDF is parallel to each base and why its length is one half thesum of the lengths of the bases.

USING TECHNOLOGY In Exercises 43–45, use geometry software.Draw points A, B, C and segments AC

Æ

and BCÆ

. Construct a circle with center A and radius AC. Construct a circle with center B and radius BC. Label the other intersection of the circles D. Draw BD

Æand AD

Æ.

43. What kind of shape is ACBD? How doyou know? What happens to the shapeas you drag A? drag B? drag C?

44. Measure ™ACB and ™ADB. What happensto the angle measures as you drag A, B, or C?

45. Which theorem does this construction illustrate?

A

B

C D

E

F

A B

CD E

T S

R

U

q

D

B

C

A

A F

E

D

B

C

G



INTE

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STUDENT HELP

CAKE DESIGNERSdesign cakes for

many occasions, includingweddings, birthdays,anniversaries, andgraduations.


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FOCUS ONCAREERS


46. PROVING THEOREM 6.18 Write a two-column proof of Theorem 6.18.

GIVEN � ABÆ

£ CBÆ

, ADÆ

£ CDÆ

PROVE � ACÆ

fi BDÆ

47. PROVING THEOREM 6.19 Write a paragraph proof of Theorem 6.19.

GIVEN � ABCD is a kite with ABÆ

£ CBÆ

and ADÆ

£ CDÆ.

PROVE � ™A £ ™C, ™B ‹ ™D

Plan for Proof First show that ™A £ ™C. Then use an indirect argumentto show ™B ‹ ™D: If ™B £ ™D, then ABCD is a parallelogram. Butopposite sides of a parallelogram are congruent. This contradicts thedefinition of a kite.

TRAPEZOIDS Decide whether you are given enough information to conclude that ABCD is an isosceles trapezoid. Explain your reasoning.

48. ABÆ

∞ DCÆ

49. ABÆ

∞ DCÆ

50. ™A £ ™B

ADÆ

£ BCÆ

ACÆ

£ BDÆ

™D £ ™C

ADÆ

£ ABÆ

™A ‹ ™C ™A ‹ ™C

51. MULTIPLE CHOICE In the trapezoid at theright, NP = 15. What is the value of x?

¡A 2 ¡B 3 ¡C 4

¡D 5 ¡E 6

52. MULTIPLE CHOICE Which one of the following can a trapezoid have?

¡A congruent bases

¡B diagonals that bisect each other

¡C exactly two congruent sides

¡D a pair of congruent opposite angles

¡E exactly three congruent angles

53. PROOF Prove one direction of Theorem 6.16: If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.

GIVEN � PQRS is a trapezoid.

QRÆ

∞ PSÆ

, PRÆ

£ SQÆ

PROVE � QPÆ

£ RSÆ

Plan for Proof Draw a perpendicular segment from Q to PSÆ

and label theintersection M. Draw a perpendicular segment from R to PS

Æand label the

intersection N. Prove that ¤QMS £ ¤RNP. Then prove that ¤QPS £ ¤RSP.

A

B

C

DX

A

B

C

D

A B

CD

K L

MJ

N P

2x � 2

3x � 2

15

R

P S

q

TestPreparation

★★ Challenge

EXTRA CHALLENGE



CONDITIONAL STATEMENTS Rewrite the statement in if-then form. (Review 2.1)

54. A scalene triangle has no congruent sides.

55. A kite has perpendicular diagonals.

56. A polygon is a pentagon if it has five sides.

FINDING MEASUREMENTS Use the diagram to findthe side length or angle measure. (Review 6.2 for 6.6)

57. LN 58. KL 59. ML

60. JL 61. m™JML 62. m™MJK

PARALLELOGRAMS Determine whether the given points represent thevertices of a parallelogram. Explain your answer. (Review 6.3 for 6.6)

63. A(º2, 8), B(5, 8), C(2, 0), D(º5, 0)

64. P(4, º3), Q(9, º1), R(8, º6), S(3, º8)

1. POSITIONING BUTTONS The tool at the rightis used to decide where to put buttons on a shirt. Thetool is stretched to fit the length of the shirt, and thepointers show where to put the buttons. Why are thepointers always evenly spaced? (Hint: You can provethat HJ

Æ£ JK

Æif you know that ¤JFK £ ¤HEJ.)

(Lesson 6.4)

Determine whether the given points representthe vertices of a rectangle, a rhombus, asquare, a trapezoid, or a kite. (Lessons 6.4, 6.5)

2. P(2, 5), Q(º4, 5), R(2, º7), S(º4, º7)

3. A(º3, 6), B(0, 9), C(3, 6), D(0, º10)

4. J(º5, 6), K(º4, º2), L(4, º1), M(3, 7)

5. P(º5, º3), Q(1, º2), R(6, 3), S(7, 9)

6. PROVING THEOREM 6.15 Write a proof of Theorem 6.15.

GIVEN � ABCD is a trapezoid with ABÆ

∞ DCÆ

.™D £ ™C

PROVE � ADÆ

£ BCÆ

Plan for Proof Draw AEÆ

so ABCE is a parallelogram. Use the TransitiveProperty of Congruence to show ™AED £ ™D. Then AD

Æ£ AE

Æ, so

ADÆ

£ BCÆ

. (Lesson 6.5)

QUIZ 2 Self-Test for Lessons 6.4 and 6.5

MIXED REVIEW

M

K L

J 10

100�7

5.6

N

A B

CD E

A

B

C

D

E

F

G

H

J

K

6.6 Special Quadrilaterals 367

1. In Example 2, explain how to prove that EFÆ

∞ HGÆ

.

Copy the chart. Put an X in the box if the shape always has the given property.

2.

3.

4.

5.

6.

7. Which quadrilaterals can you form with four sticks of the same length? Youmust attach the sticks at their ends and cannot bend or break any of them.

PROPERTIES OF QUADRILATERALS Copy the chart. Put an X in the box ifthe shape always has the given property.

8.

9.

10.

11.

12.

13.

TENT SHAPES What kind of special quadrilateral is the red shape?

14. 15.


GUIDED PRACTICE

Property ⁄ Rectangle Rhombus Square Kite Trapezoid

Both pairs of opp. sides are ∞. ? ? ? ? ? ?

Exactly 1 pair of ? ? ? ? ? ?opp. sides are ∞.

Diagonals are fi. ? ? ? ? ? ?

Diagonals are £. ? ? ? ? ? ?

Diagonals bisect each other. ? ? ? ? ? ?

Property ⁄ Rectangle Rhombus Square Kite Trapezoid

Both pairs of opp. sides are £. ? ? ? ? ? ?

Exactly 1 pair of ? ? ? ? ? ?opp. sides are £.

All sides are £. ? ? ? ? ? ?

Both pairs of opp.√ are £.

? ? ? ? ? ?

Exactly 1 pair of opp. ? ? ? ? ? ?√ are £.

All √ are £. ? ? ? ? ? ?


STUDENT HELP

Skill Check ✓Concept Check ✓

Tents are designeddifferently for

different climates. Forexample, winter tents aredesigned to shed snow.Desert tents can have flatroofs because they don’tneed to shed rain.

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IDENTIFYING QUADRILATERALS Identify the special quadrilateral. Use themost specific name.

16. 17. 18.

IDENTIFYING QUADRILATERALS What kinds of quadrilaterals meet theconditions shown? ABCD is not drawn to scale.

19. 20. 21.

22. 23. 24.

DESCRIBING METHODS OF PROOF Summarize the ways you have learnedto prove that a quadrilateral is the given special type of quadrilateral.

25. kite 26. square 27. rectangle

28. trapezoid 29. isosceles trapezoid

DEVELOPING PROOF Which two segments or angles must becongruent to enable you to prove ABCD is the given quadrilateral? Explainyour reasoning. There may be more than one right answer.

30. isosceles trapezoid 31. parallelogram 32. rhombus

33. rectangle 34. kite 35. square

QUADRILATERALS What kind of quadrilateral is PQRS? Justify your answer.

36. P(0, 0), Q(0, 2), R(5, 5), S(2, 0) 37. P(1, 1), Q(5, 1), R(4, 8), S(2, 8)

38. P(2, 1), Q(7, 1), R(7, 7), S(2, 5) 39. P(0, 7), Q(4, 8), R(5, 2), S(1, 1)

40. P(1, 7), Q(5, 9), R(8, 3), S(4, 1) 41. P(5, 1), Q(9, 6), R(5, 11), S(1, 6)

A

B C

D

B

CA

DA

E

D

B C

A

B

E

D

C

A

B

D

C

E

A

B

D

C110�

70�

AB

DC

AB

DC

A B

D C

A B

D C

B

A

D

CB

A

D

C

STUDENT HELP

Study TipSee the summaries forparallelograms andrhombuses on pp. 340and 365, and the list ofpostulates and theoremson pp. 828–837. You canrefer to your summariesas you do the rest of theexercises.

STUDENT HELP


30–35Example 2: Exs. 14–18,

42–44Example 3: Exs. 25–29,

36–41Example 4: Exs. 14–18,

42, 43Example 5: Exs. 45–47

6.6 Special Quadrilaterals 369

GEM CUTTING In Exercises 42 and 43, use the following information.There are different ways of cutting gems toenhance the beauty of the jewel. One of theearliest shapes used for diamonds is called the table cut, as shown at the right. Each face of a cut gem is called a facet.

42. BCÆ

∞ ADÆ

, ABÆ

and DCÆ

are not parallel.What shape is the facet labeled ABCD?

43. DEÆ

∞ GFÆ

, DGÆ

and EFÆ

are congruent, but not parallel. What shape is the facetlabeled DEFG?

44. JUSTIFYING A CONSTRUCTION Look back at the Perpendicular to a Lineconstruction on page 130. Explain why this construction works.

DRAWING QUADRILATERALS Draw ACÆ

and BDÆ

as described. What specialtype of quadrilateral is ABCD? Prove that your answer is correct.

45. ACÆ

and BDÆ

bisect each other, but they are not perpendicular or congruent.

46. ACÆ

and BDÆ

bisect each other. ACÆ

fi BDÆ

, ACÆ

‹ BDÆ

47. ACÆ

fi BDÆ

, and ACÆ

bisects BDÆ

. BDÆ

does not bisect ACÆ

.

48. LOGICAL REASONING EFGH, GHJK, and JKLM are all parallelograms.If EF

Æand LM

Æare not collinear, what kind of quadrilateral is EFLM? Prove

that your answer is correct.

49. PROOF Prove that the median of a righttriangle is one half the length of the hypotenuse.

GIVEN � ™CDE is a right angle. CMÆ

£ EMÆ

PROVE � DMÆ

£ CMÆ

Plan for Proof First draw CFÆ

and EFÆ

so CDEF is a rectangle. (How?)

50. PROOF Use facts about angles to prove that the quadrilateral in Example5 is a rectangle. (Hint: Let x° be the measure of ™ABN. Find the measures ofthe other angles in terms of x.)

PROOF What special type of quadrilateral is EFGH? Prove that youranswer is correct.

GEMOLOGISTSanalyze the cut of

a gem when determining its value.


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www.mcdougallittell.comfor help with Exs. 45–47.

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STUDENT HELP

51. GIVEN � PQRS is a square.E, F, G, and H aremidpoints of the sidesof the square.

52. GIVEN � JKÆ

£ LMÆ

, E, F, G, andH are the midpoints ofJLÆ

, KLÆ

, KMÆ

, and JMÆ

.

BA

C

D

E

GF

D

C F

E

M

F

EH

G

J

L

M

KP F q

E

S H

G

R


53. MULTI-STEP PROBLEM Copy the diagram. JKLMNis a regular pentagon. You will identify JPMN.

a. What kind of triangle is ¤JKL? Use ¤JKL toprove that ™LJN £ ™JLM.

b. List everything you know about the interior anglesof JLMN. Use these facts to prove that JL

Æ∞ NMÆ

.

c. Reasoning similar to parts (a) and (b) shows that KMÆ

∞ JNÆ

. Based on thisand the result from part (b), what kind of shape is JPMN?

d. Writing Is JPMN a rhombus? Justify your answer.

54. PROOF ACÆ

and BDÆ

intersect each other at N. ANÆ

£ BNÆ

and CNÆ

£ DNÆ

,but AC

Æand BD

Ædo not bisect each other. Draw AC

Æand BD

Æ, and ABCD.

What special type of quadrilateral is ABCD? Write a plan for a proof of your answer.

FINDING AREA Find the area of the figure. (Review 1.7 for 6.7)

55. 56. 57.

58. 59. 60.

USING ALGEBRA In Exercises 61 and 62, use thediagram at the right. (Review 6.1)

61. What is the value of x?

62. What is m™A? Use your result from Exercise 61.

FINDING THE MIDSEGMENT Find the length of the midsegment of thetrapezoid. (Review 6.5 for 6.7)

63. 64. 65.

2

y

xA

B

DC

�42

y

x

AB

DC

6

2

2

y

x

A B

D C

xyxy

12

1085

12

136

9

3

5

74

4

4

4

MIXED REVIEW

TestPreparation

★★ Challenge

J

LM

K

P

N

A

B

D C

(32x � 15)�

133�

80� (44x � 1)�


1. What is the midsegment of a trapezoid?

2. If you use AB as the base to find thearea of ⁄ABCD shown at the right,what should you use as the height?

Match the region with a formula for its area. Use each formula exactly once.

3. Region 1 ¡A A = s2

4. Region 2 ¡B A = �12�d1d2

5. Region 3 ¡C A = �12�bh

6. Region 4 ¡D A = �12�h(b1 + b2)

7. Region 5 ¡E A = bh

Find the area of the polygon.

8. 9. 10.

11. 12. 13.

FINDING AREA Find the area of the polygon.

14. 15. 16.

17. 18. 19.

5

97

5 7


94

5

7

4

GUIDED PRACTICE

6

B

A

C

5

E

F

D7.5

4

3

5

241

Vocabulary Check ✓Concept Check ✓

Skill Check ✓


STUDENT HELP

15

8

22

21 4

75

5 5

4

42 10

6

6

6

8

4

STUDENT HELP


41–47Example 2: Exs. 26–31

continued on p. 377

6.7 Areas of Triangles and Quadrilaterals 377

FINDING AREA Find the area of the polygon.

20. 21. 22.

23. 24. 25.


26. A = 63 cm2 27. A = 48 ft2 28. A = 48 in.2

REWRITING FORMULAS Rewrite the formula for the area of the polygon interms of the given variable. Use the formulas on pages 372 and 374.

29. triangle, b 30. kite, d1 31. trapezoid, b1

FINDING AREA Find the area of quadrilateral ABCD.

32. 33. 34.

ENERGY CONSERVATION The total area of a building’s windows affectsthe cost of heating or cooling the building. Find the area of the window.

35. 36.

37. 38.

12 in.

48 in.

32 in.3 ft

2 ft

y

x

A

D

B

C

1

3

y

x

1

A

D

B

C

3

y

x

1

2A D

B C

8 in.x

8 in.

xx

4 ft

4 ftx

7 cm

xyxy

7 75

5

16

14

816

15

24

7

2419

38

8

6

10

20 in.

21 in.

9 in.

16 in.12 in.

16 in.

16 in.

30 in.12 in.

INSULATIONInsulation makes a

building more energyefficient. The ability of amaterial to insulate is calledits R-value. Many windowshave an R-value of 1. Adobehas an R-value of 11.9.

APPLICATION LINKwww.mcdougallittell.com

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STUDENT HELP

HOMEWORK HELPcontinued from p. 376

Example 3: Exs. 26–28, 39, 40

Example 4: Exs. 32–34Example 5: Exs. 20–25,

44Example 6: Exs. 35–38,

48–52


39. LOGICAL REASONING Are all parallelograms with an area of 24 squarefeet and a base of 6 feet congruent? Explain.

40. LOGICAL REASONING Are all rectangles with an area of 24 square feetand a base of 6 feet congruent? Explain.

USING THE PYTHAGOREAN THEOREM Find the area of the polygon.

41. 42. 43.

44. LOGICAL REASONING What happens to the area of a kite if you doublethe length of one of the diagonals? if you double the length of bothdiagonals?

PARADE FLOATS You are decorating a float for a parade. You estimatethat, on average, a carnation will cover 3 square inches, a daisy will cover 2 square inches, and a chrysanthemum will cover 4 square inches. Abouthow many flowers do you need to cover the shape on the float?

45. Carnations: 2 ft by 5 ft rectangle

46. Daisies: trapezoid (b1 = 5 ft, b2 = 3 ft, h = 2 ft)

47. Chrysanthemums: triangle (b = 3 ft, h = 8 ft)

BRIDGES In Exercises 48 and 49, use the following information.The town of Elizabethton, Tennessee, restored the roof of this covered bridge with cedar shakes, a kind of rough woodenshingle. The shakes vary in width, but theaverage width is about 10 inches. So, on average, each shake protects a 10 inch by 10 inch square of roof.

48. In the diagram of the roof, the hidden back and left sides are the same as thefront and right sides. What is the total area of the roof?

49. Estimate the number of shakes needed to cover the roof.

AREAS Find the areas of the blue and yellow regions.

50. 51. 52. 84�2

7�2 14

12

8

8

8

83

6

6

3

6 3

3 6

16

208

13

126 10

STUDENT HELP

Study TipRemember that twopolygons are congruentif their correspondingangles and sides arecongruent.

137 ft

159 ft21 ft

13 ft 2 in.

13 ft 7 in.

MARKCANDELARIA

When Mark Candelariarestored historic buildings inScottsdale, Arizona, hecalculated the areas of thewalls and floors that neededto be replaced.

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FOCUS ONPEOPLE

6.7 Areas of Triangles and Quadrilaterals 379

JUSTIFYING THEOREM 6.20 In Exercises 53–57, you will justify theformula for the area of a rectangle. In the diagram, AEJH and JFCG arecongruent rectangles with base length b and height h.

53. What kind of shape is EBFJ? HJGD? Explain.

54. What kind of shape is ABCD? How do you know?

55. Write an expression for the length of a side of ABCD.Then write an expression for the area of ABCD.

56. Write expressions for the areas of EBFJ and HJGD.

57. Substitute your answers from Exercises 55 and 56 into the following equation.

Let A = the area of AEJH. Solve the equation to find an expression for A.Area of ABCD = Area of HJGD + Area of EBFJ + 2(Area of AEJH)

JUSTIFYING THEOREM 6.23 Exercises 58 and 59 illustrate two ways toprove Theorem 6.23. Use the diagram to write a plan for a proof.

58. GIVEN � LPQK is a trapezoid as 59. GIVEN � ABCD is a trapezoid asshown. LPQK £ PLMN. shown. EBCF £ GHDF.

PROVE � The area of LPQK is PROVE � The area of ABCD is

�12�h(b1 + b2). �

12�h(b1 + b2).

60. MULTIPLE CHOICE What is the area of trapezoid EFGH?

¡A 25 in.2 ¡B 416 in.2

¡C 84 in.2 ¡D 42 in.2

¡E 68 in.2

61. MULTIPLE CHOICE What is the area of parallelogram JKLM?

¡A 12 cm2 ¡B 15 cm2

¡C 18 cm2 ¡D 30 cm2

¡E 40 cm2

62. Writing Explain why the area of anyquadrilateral with perpendicular

diagonals is A = �12�d1d2, where d1 and d2

are the lengths of the diagonals.

H

G

h

Ch

b

F

b

A

D

BE

J

K b2 M

q b2

L

P

h

b1

b1

N

B b1

G

A b2

C

D H

h FE

4 in.

13 in.E H

8 in.F G

J

d2S

q

TP Rd1

Look Back For help with squaringbinomial expressions, seep. 798.

STUDENT HELP

TestPreparation

★★ Challenge

3 cm

M

5 cmK L

J

EXTRA CHALLENGE



CLASSIFYING ANGLES State whether the angle appears to be acute, right,or obtuse. Then estimate its measure. (Review 1.4 for 7.1)

63. 64. 65.

PLACING FIGURES IN A COORDINATE PLANE Place the triangle in acoordinate plane and label the coordinates of the vertices. (Review 4.7 for 7.1)

66. A triangle has a base length of 3 units and a height of 4 units.

67. An isosceles triangle has a base length of 10 units and a height of 5 units.

USING ALGEBRA In Exercises 68–70, AEÆ

, BFÆ

, and CGÆ

are medians. Findthe value of x. (Review 5.3)

68. 69. 70.

What special type of quadrilateral is shown? Give the most specific name,and justify your answer. (Lesson 6.6)

1. 2. 3.

The shape has an area of 60 square inches. Find the value of x. (Lesson 6.7)

4. 5. 6.

7. GOLD BULLION Gold bullion is molded intoblocks with cross sections that are isoscelestrapezoids. A cross section of a 25 kilogram blockhas a height of 5.4 centimeters and bases of 8.3 centimeters and 11 centimeters. What is the area of the cross section? (Lesson 6.7)

y

x

1

Z

1 W

Y

X

y

x

1

œ

1 S

T

R

y

x

N

M

P 3O

3

QUIZ 3 Self-Test for Lessons 6.6 and 6.7

AD

C G

BE

x � 6x

A

F

C

G

2x

4

BE

A

F

C

Bx 14

G

E

xyxy

MIXED REVIEW

x

15 in.

x

10 in.

x

12 in.

vocabulary check what is the plural of vertex · 326 chapter 6 quadrilaterals convex or concave use...

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