continuity of learning assignments essential activity ... · continuity of learning assignments...

Continuity of Learning Assignments

Grade: 9 - 12 Subject: Geometry

Days 1-2 Days 3-4 Days 5-6 Days 7-8 Days 9-10 Essential Activity Essential Activity Essential Activity Essential Activity Essential Activity

Digital Learning: https://bit.ly/3bvjyFF https://bit.ly/3bzBwXG https://bit.ly/3dBcMQC

On Paper: Textbook Notes pg. 1-4

• Section 6-1 Practice pg. 5-8

• Introduction to Polygons

• 6-1 Practice:

Digital Learning:

On Paper: Textbook Notes pg. 9-12

• Section 6-2: Practice pg.13-16

• Polygon Properties

• 6-2 Practice

Digital Learning: https://bit.ly/3dAluP5 https://bit.ly/3bvjNk3 https://bit.ly/3avbrZC https://bit.ly/39saAYc https://bit.ly/2wyfKop On Paper: Textbook Notes pg. 17-24

• Section 6-5/6-6 Practice pg. 25

• Complete Quadrilateral Graphic Organizer

Digital Learning: https://bit.ly/3dDFtfx https://bit.ly/3bqePFg On Paper: Practice pg. 26-29

• 6-5 Practice • Polygon

Practice: Properties of Parallelograms

Digital Learning:

On Paper: Practice pg. 30-33

• 6-6 Practice • Special

Quadrilaterals Practice

Extension Extension Extension Extension Extension Digital Learning Mastery Prep Khan Academy: Geometry

Digital Learning Mastery Prep Khan Academy: Geometry

Digital Learning

Mastery Prep Khan Academy: Geometry

Digital Learning


Digital Learning


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Vocabulary

Review

Chapter 6 146

6-1 The Polygon Angle-Sum Theorems

1. Underline the correct word to complete the sentence.

In a convex polygon, no point on the lines containing the sides of the polygon is in the

interior / exterior of the polygon.

2. Cross out the polygon that is NOT convex.

Vocabulary Builder

regular polygon (noun) REG yuh lur PAHL ih gahn

Definition: A regular polygon is a polygon that is both equilateral and equiangular.

Example: An equilateral triangle is a regular polygon with three congruent sides and three congruent angles.

Use Your VocabularyUnderline the correct word(s) to complete each sentence.

3. The sides of a regular polygon are congruent / scalene .

4. A right triangle is / is not a regular polygon.

5. An isosceles triangle is / is not always a regular polygon.

Write equiangular, equilateral, or regular to identify each hexagon. Use each word once.

6. 7.

120í120í

120í

120í

120í120í 8. 120í120í

120í

120í120í

120í

equilateral regular equiangular

HSM11GEMC_0601.indd 146HSM11GEMC_0601.indd 146 3/8/09 3:42:58 PM3/8/09 3:42:58 PM

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Problem 1

Theorem 6-1 Polygon Angle-Sum Theorem and Corollary

A

Problem 2

147 Lesson 6-1

Finding a Polygon Angle Sum

Got It? What is the sum of the interior angle measures of a 17-gon?

11. Use the justifications below to find the sum. 12. Draw diagonals from vertex A to check your answer.

sum 5 Q 2 2 R180 Polygon Angle-Sum Theorem

5 Q 2 2 R180 Substitute for n.

5 ? 180 Subtract.

5 Simplify.

13. The sum of the interior angle measures of a 17-gon is .

Theorem 6-1 Th e sum of the measures of the interior angles of an n-gon is (n 2 2)180.

Corollary Th e measure of each interior angle of a regular n-gon is (n 2 2)180

n .

9. When n 2 2 5 1, the polygon is a(n) 9.

10. When n 2 2 5 2, the polygon is a(n) 9.

Using the Polygon Angle-Sum Theorem

Got It? What is the measure of each interior angle in a regular nonagon?

Underline the correct word or number to complete each sentence.

14. The interior angles in a regular polygon are congruent / different .

15. A regular nonagon has 7 / 8 / 9 congruent sides.

16. Use the Corollary to the Polygon Angle-Sum Theorem to find the measure of each interior angle in a regular nonagon.

Measure of an angle 5Q 2 2 R180

5Q R180

5

17. The measure of each interior angle in a regular nonagon is .

triangle

quadrilateral

17

9

7

9

140

140

9

15

2700

2700

n


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Problem 3

Problem 4

Theorem 6-2 Polygon Exterior Angle-Sum Theorem

E

F

G

H

120�

85� 53�

1

23

4

5

Chapter 6 148

Using the Polygon Angle-Sum Theorem

Got It? What is mlG in quadrilateral EFGH?

18. Use the Polygon Angle-Sum Theorem to find m/G for n 5 4.

m/E 1 m/F 1 m/G 1 m/H 5 (n 2 2)180

m/E 1 m/F 1 m/G 1 m/H 5 Q 2 2 R180

1 1 1 5 ? 180

m/G 1 5

m/G 5

19. m/G in quadrilateral EFGH is .

Finding an Exterior Angle Measure

Got It? What is the measure of an exterior angle of a regular nonagon?

Underline the correct number or word to complete each sentence.

23. Since the nonagon is regular, its interior angles are congruent / right .

24. The exterior angles are complements / supplements of the interior angles.

25. Since the nonagon is regular, its exterior angles are congruent / right .

26. The sum of the measures of the exterior angles of a polygon is 180 / 360 .

27. A regular nonagon has 7 / 9 / 12 sides.

Th e sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

20. In the pentagon below, m/1 1 m/2 1 m/3 1 m/4 1 m/5 5 .

Use the Polygon Exterior Angle-Sum Theorem to find each measure.

21. 81í

72í87í

120í 22.

75í

56í90í

66í73í

120 1 81 1 1 87 5 360 90 1 1 75 1 73 1 66 5

4

85 120 m/G 53

258

2

360

102

102

360

72 56 360


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Lesson Check

Math Success

Now Iget it!

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0 2 4 6 8 10

149 Lesson 6-1

28. What is the measure of an exterior angle of a regular nonagon? Explain.

_______________________________________________________________________

_______________________________________________________________________

• Do you UNDERSTAND?

Error Analysis Your friend says that she measured an interior angle of a regular polygon as 130. Explain why this result is impossible.

29. Use indirect reasoning to find a contradiction.

Assume temporarily that a regular n-gon has a 1308 interior angle.

angle sum 5 ? n A regular n-gon has n congruent angles.

angle sum 5 Q R180 Polygon Angle-Sum Theorem

5 Q R180 Use the Transitive Property of Equality.

5 2 Use the Distributive Property.

5 Subtract 180n from each side.

n 5 Divide each side by 250.

n 2 The number of sides in a polygon is a whole number $ 3.

30. Explain why your friend’s result is impossible.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Check off the vocabulary words that you understand.

equilateral polygon equiangular polygon regular polygon

Rate how well you can fi nd angle measures of polygons.

Answers may vary. Sample: If each interior angle measures 130,

then the regular polygon would have 7.2 sides. This is impossible

because the number of sides is an integer greater than 2. So, each

angle cannot be 130.

40°. Explanations may vary. Sample: Divide 360 by 9, the number

of sides in a nonagon.

130

360

n 2 2

n 2 2

180n

2360

130n

130n

250n

7.2

7.2


©9 j2p0l1i1c GKTu9taan JS6oVf3tZw5aYrDeV 4LYLvCX.E N mAQlilk 8r2ipg3hDttsU irjehs4eTr1vheNdJ.r s NM1aqd5eI cwuixtahH 7IBnef8iKniiuteep BGreaoYmPewt0rKyZ.x Worksheet by Kuta Software LLC

Kuta Software - Infinite Geometry Name___________________________________

Period____Date________________Introduction to Polygons

Write the name of each polygon.

1) 2)

3) 4)

5) 6)

7) 8)

State if each polygon is concave or convex.

9) 10)

-1-

©Y S2R0O1c1Y xKDuItOaW 5Soo4fbt2wtaRrEef qL1LPC5.m c EAhlDlg MryivgkhZtxsM yraegsQe3r7vAeWdD.1 r NMKatdce9 awkiut2hz BIrnOf7i9nBiStQei CGWedoVmse8tjrRy3.j Worksheet by Kuta Software LLC

11) 12)

13) 14)

State if each polygon is regular or not.

15) 16)

17) 18)

19) 20)

Critical thinking questions:

21) Sketch a concave hexagon. 22) Which are impossible: Regular convex octagon Concave trapezoid Convex irregular 20-gon Concave triangle Concave equilateral pentagon

-2-

Name Class Date

6-1 Practice Form G

1. 2. 3.

4. 12-gon 5. 18-gon 6. 25-gon

7. 60-gon 8. 102-gon 9. 17-gon

10. 36-gon 11. 90-gon 12. 11-gon

Find the measure of one angle in each regular polygon. Round to the nearest tenth if necessary.

13. 14. 15.

16. regular 15-gon 17. regular 11-gon 18. regular 13-gon



Algebra Find the missing angle measures.

25. 26.

27. 28. 29.

Prentice Hall Gold Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

3

The Polygon Angle-Sum Theorems

Find the sum of the angle measures of each polygon.

Name

Class

Date

6-1 Practice (continued) Form G

30. 31. 32.

33. 34. 35.

Find the measure of an exterior angle of each regular polygon. Round to the nearest tenth if necessary.

36. decagon 37. 16-gon 38. hexagon

39. 20-gon 40. 72-gon 41. square

42. 15-gon 43. 25-gon 44. 80-gon

Find the values of the variables for each regular polygon. Round to the nearest tenth if necessary.

45. 46. 47.

48. Reasoning Can a quadrilateral have no obtuse angles? Explain.

The measure of an exterior angle of a regular polygon is given. Find the measure of an interior angle. Then find the number of sides.

49. 12 50. 6 51. 45

52. 40 53. 24 54. 18

55. 9 56. 14.4 57. 7.2


4

The Polygon Angle-Sum Theorems

Algebra Find the missing angle measures.

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Vocabulary

Review

Chapter 6 150

Properties of Parallelograms6-2

1. Supplementary angles are two angles whose measures sum to .

2. Suppose /X and /Y are supplementary. If m/X 5 75, then m/Y 5 .

Underline the correct word to complete each sentence.

3. A linear pair is complementary / supplementary .

4. /AFB and /EFD at the right are complementary / supplementary .

Vocabulary Builder

consecutive (adjective) kun SEK yoo tiv

Definition: Consecutive items follow one after another in uninterrupted order.

Math Usage: Consecutive angles of a polygon share a common side.

Examples: The numbers 23, 22, 21, 0, 1, 2, 3, . . . are consecutive integers.

Non-Example: The letters A, B, C, F, P, . . . are NOT consecutive letters of the alphabet.

Use Your VocabularyUse the diagram at the right. Draw a line from each angle in Column A to a consecutive angle in Column B.

Column A Column B

5. /A /F

6. /C /E

7. /D /D

Write the next two consecutive months in each sequence.

8. January, February, March, April,

,

9. December, November, October, September,

,

120í

60í

D

F

CBA

E

CF

A B

DE

May

August July

June

180

105


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Problem 1

Problem 2

Theorems 6-3, 6-4, 6-5, 6-6

B

EA D

C

S

P

R

Q

64�

151 Lesson 6-2

Using Consecutive Angles

Got It? Suppose you adjust the lamp so that mlS is 86. What is mlR in ~PQRS?

Underline the correct word or number to complete each statement.

13. /R and /S are adjacent / consecutive angles, so they are supplementary.

14. m/R 1 m/S 5 90 / 180

15. Now find m/R.

16. m/R 5 .

Using Properties of Parallelograms in a Proof

Got It? Use the diagram at the right.

Given: ~ABCD, AK > MK Prove: /BCD > /CMD

17. Circle the classification of nAKM .

equilateral isosceles right

18. Complete the proof. The reasons are given.

Statements Reasons

1) AK > 1) Given

2) /DAB > 2) Angles opposite congruent sides of a triangle are congruent.

3) /BCD > 3) Opposite angles of a parallelogram are congruent.

4) /BCD > 4) Transitive Property of Congruence

Theorem 6-3 If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Theorem 6-4 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

Theorem 6-5 If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Theorem 6-6 If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Use the diagram at the right for Exercises 10–12.

10. Mark parallelogram ABCD to model Theorem 6-3 and Theorem 6-5.

11. AE > 12. BE >

K

CB

A MD

CE

MK

lCMD

lDAB

lCMD

94

DE

mlR 1 86 5 180 mlR 5 180 2 86 mlR 5 94


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Problem 3

Theorem 6-7

Q

ST

R

P

x � 1 2x

3y � 7 y

A B

C D

E F

Chapter 6 152

Using Algebra to Find Lengths

Got It? Find the values of x and y in ~PQRS at the right. What are PR and SQ?

19. Circle the reason PT > TR and ST > TQ.

Diagonals of a Opposite sides of PR is the parallelogram a parallelogram perpendicular bisect each other. are congruent. bisector of QS.

20. Cross out the equation that is NOT true.

3(x 1 1) 2 7 5 2x y 5 x 1 1 3y 2 7 5 x 1 1 3y 2 7 5 2x

21. Find the value of x. 22. Find the value of y.

23. Find PT. 24. Find ST.

PT 5 3 2 7 ST 5 1 1

PT 5 2 7 ST 5

PT 5

25. Find PR. 26. Find SQ.

PR 5 2( ) SQ 5 2( )

PR 5 SQ 5

27. Explain why you do not need to find TR and TQ after finding PT and ST.

_______________________________________________________________________

_______________________________________________________________________

If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Use the diagram at the right for Exercises 28 and 29.

28. If *AB)6*CD)6*EF) and AC > CE , then BD > .

29. Mark the diagram to show your answer to Exercise 28.

3(x 1 1) 2 7 5 2x 3x 1 3 2 7 5 2x 3x2 4 5 2x x5 4

y5 x 1 1

y5 4 1 1y5 5

5

5

5

8

8

15

16 10

DF

4

Answers may vary. Sample: The diagonals of a parallelogram

bisect each other, so PT 5 TR and ST 5 TQ.

HSM11GEMC_0602.indd 152HSM11GEMC_0602.indd 152 3/17/09 9:13:20 AM3/17/09 9:13:20 AM

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Lesson Check

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0 2 4 6 8 10

Math Success

Problem 4

153 Lesson 6-2


Error Analysis Your classmate says that QV 5 10. Explain why the statement may not be correct.

35. Place a ✓ in the box if you are given the information. Place an ✗ if you are not given the information.

three lines cut by two transversals

three parallel lines cut by two transversals

congruent segments on one transversal

36. What needs to be true for QV to equal 10?

_______________________________________________________________________

37. Explain why your classmate’s statement may not be correct.

_______________________________________________________________________


parallelogram opposite sides opposite angles consecutive angles

Rate how well you understand parallelograms.

Using Parallel lines and Transversals

Got It? In the figure at the right, *AE) n*BF)n*CG) n*DH). If EF 5 FG 5 GH 5 6

and AD 5 15, what is CD?

30. You know that the parallel lines cut off congruent segments on transversal .

31. By Theorem 6-7, the parallel lines also cut off congruent segments on .

32. AD 5 AB 1 BC 1 by the Segment Addition Postulate.

33. AB 5 5 CD, so AD 5 ? CD. Then CD 5 ? AD.

34. You know that AD 5 15, so CD 5 ? 15 5 .

A

EF

GH

B C D

VS

QR

T

P5 cm

✓

13

13

3

5

*EH)

*AD)

CD

BC

✓

✗

The lines cut by transversals need to be parallel.

Answers may vary. Sample: The lines may not be parallel.


_____POLYGONPROPERTIES NAME:_____________________________________________________

BONUS:Ifthemeasureofoneinteriorangleofaregularpolygonis168o,howmanysidesdoesthepolygonhave?

Fornumbers1-4usethefiguretodeterminex.1.

2.

3.

4.

5. Fullyclassifythepolygoninnumber4.

Fornumbers6-7,usethefiguretodeterminetheindicatedquantities.

6.

7.

8. Whatisthemeasureofoneinteriorangleofaregulardodecagon?

9. Whatisthemeasureofoneexteriorangleofaregularnonagon?

10. Ifthemeasureofoneexteriorangleofaregularpolygonis24o,howmanysidesdoesthepolygonhave?

Fornumbers1-4usethefiguretodeterminex.

BONUS:Ifthemeasureofaninteriorangleofaregularpolygonis144o,howmanysidesdoesthepolygonhave?

1.

2.

3.

4.

5. Fullyclassifythepolygoninnumber4.

Fornumber6-7,usethefiguretodeterminetheindicatedquantities.

6.

7.

8. Determinethemeasureofoneinteriorangleonaregularhendecagon.

9. Determinethemeasureofoneexteriorangleofaregularpentagon.

10. Ifthemeasureofoneexteriorangleofaregularpolygonis40o,howmanysidesdoesthepolygonhave?

Name Class Date

6-2 Practice Form G

Find the value of x in each parallelogram. 1. 2. 3. 4. 5. 6.

Developing Proof Complete this two-column proof.

7. Given: EFGH, with diagonals and

Prove: ∆EFK @ ∆GHK

Statements Reasons

1)

2)

3)

4)

1) Given 2) The diagonals of a parallelogram

bisect each other. 3)

4)

Algebra Find the values for x and y in ABCD.

8. AE = 3x, EC = y, DE = 4x, EB = y + 1

9. AE = x + 5, EC = y, DE = 2x + 3, EB = y + 2

10. AE = 3x, EC = 2y - 2, DE = 5x, EB = 2y + 2

11. AE = 2x, EC = y + 4, DE = x, EB = 2y - 1

12. AE = 4x, EC = 5y - 2, DE = 2x, EB = y + 14


13

EGHF

EF GH@

Properties of Parallelograms

Name Class Date

6-2 Practice (continued) Form G

In the figure, TU = UV. Find each length.

13. NM 14. QR

15. LN 16. QS

Find the measures of the numbered angles for each parallelogram.

17. 18.

19. 20.

21. 22.

23. 24.

25. Developing Proof A rhombus is a parallelogram with four congruent sides. Write a plan for the following proof that uses SSS and a property of parallelograms. Given: Rhombus ABCD with diagonals and

intersecting at E

Prove:


14

ACBD

AC BD^


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Review

Vocabulary

Chapter 6 162

6-5 Conditions for Rhombuses, Rectangles, and Squares

1. A quadrilateral is a polygon with sides.

2. Cross out the figure that is NOT a quadrilateral.

Vocabulary Builder

diagonal (noun) dy AG uh nul

Definition: A diagonal is a segment with endpoints at two nonadjacent vertices of a polygon.

Word Origin: The word diagonal comes from the Greek prefix dia-, which means “through,” and gonia, which means “angle” or “corner.”

Use Your Vocabulary 3. Circle the polygon that has no diagonal.

triangle quadrilateral pentagon hexagon

4. Circle the polygon that has two diagonals.

triangle quadrilateral pentagon hexagon

5. Draw the diagonals from one vertex in each figure.

6. Write the number of diagonals you drew in each of the figures above.

pentagon: hexagon: heptagon:

,

diagonals

432

4


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d.Theorems 6-16, 6-17, and 6-18

Problem 1

A

B

D

C

34

21

A

B

D

C

163 Lesson 6-5

Theorem 6-16 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Theorem 6-17 If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

7. Insert a right angle symbol in the parallelogram at the right to illustrate Theorem 6-16. Insert congruent marks to illustrate Theorem 6-17.

Use the diagram from Exercise 7 to complete Exercises 8 and 9.

8. If ABCD is a parallelogram and AC ' , then ABCD is a rhombus.

9. If ABCD is a parallelogram, /1 > , and /3 > , then ABCD is a rhombus.

Theorem 6-18 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

10. Insert congruent marks and right angle symbols in the parallelogram to the right to illustrate Theorem 6-18.

11. Use the diagram from Exercise 10 to complete the statement.

If ABCD is a parallelogram, and BD > then ABCD is a rectangle.

12. Circle the parallelogram that has diagonals that are both perpendicular and congruent.

parallelogram rectangle rhombus square

Identifying Special Parallelograms

Got It? A parallelogram has angle measures of 20, 160, 20, and 160. Can you conclude that it is a rhombus, a rectangle, or a square? Explain.

13. Draw a parallelogram in the box below. Label the angles with their measures. Use a protractor to help you make accurate angle measurements.

160í20í20í

160í

BD

AC

l2 l4

Parallelograms may vary. Sample is given.


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Problem 2

E F

GD

5y��3 7y��5

4 4

Problem 3

Chapter 6 164

Using Properties of Special Parallelograms

Got It? For what value of y is ~DEFG a rectangle?

19. For ~DEFG to be a parallelogram, the diagonals must 9 each other.

20. EG 5 2 ( ) 21. DF 5 2 ( )

5 5

22. For ~DEFG to be a rectangle, the diagonals must be 9.

23. Now write an equation and solve for y.

24. ~DEFG is a rectangle for y 5 .

Underline the correct word or words to complete each sentence.

14. You do / do not know the lengths of the sides of the parallelogram.

15. You do / do not know the lengths of the diagonals.

16. The angles of a rectangle are all acute / obtuse / right angles.

17. The angles of a square are all acute / obtuse / right angles.

18. Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Using Properties of Parallelograms

Got It? Suppose you are on the volunteer building team at the right. You are helping to lay out a square play area. How can you use properties of diagonals to locate the four corners?

25. You can cut two pieces of rope that will be the diagonals of the square play area. Cut them the same length because a parallelogram is a 9 if the diagonals are congruent.

No. Explanations may vary. Sample: The parallelogram cannot be

a rectangle or square because it does not have four right angles.

There is not enough information to tell whether it is a rhombus.

5y 1 3

10y 1 6

7y 2 5

14y 2 10

bisect

congruent

rectangle

10y 1 6 5 14y 2 1010y 2 14y 5 210 2 6 24y 5 216 y 5 4

4


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Math Success

Lesson Check

Property Rectangle Rhombus Square

A

B

C

D

E

165 Lesson 6-5


rhombus rectangle square diagonal

Rate how well you can use properties of parallelograms.


Name all of the special parallelograms that have each property.

A. Diagonals are perpendicular. B. Diagonals are congruent. C. Diagonals are angle bisectors.

D. Diagonals bisect each other. E. Diagonals are perpendicular bisectors of each other.

29. Place a ✓ in the box if the parallelogram has the property. Place an ✗ if it does not.

26. You join the two pieces of rope at their midpoints because a quadrilateral is a 9 if the diagonals bisect each other.

27. You move so the diagonals are perpendicular because a parallelogram is a 9 if the diagonals are perpendicular.

28. Explain why the polygon is a square when you pull the ropes taut.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answers may vary. Sample: Congruent diagonals bisect each other,

so you formed a rectangle. Diagonals are perpendicular, so you

formed a rhombus. A rectangle that is a rhombus is a square.

parallelogram

rhombus


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Vocabulary

Review

Chapter 6 166

6-6 Trapezoids and Kites

Underline the correct word to complete each sentence.

1. An isosceles triangle always has two / three congruent sides.

2. An equilateral triangle is also a(n) isosceles / right triangle.

3. Cross out the length(s) that can NOT be side lengths of an isosceles triangle.

3, 4, 5 8, 8, 10 3.6, 5, 3.6 7, 11, 11

Vocabulary Builder

trapezoid (noun) TRAP ih zoyd

Related Words: base, leg

Definition: A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Main Idea: The parallel sides of a trapezoid are called bases. The nonparallel sides are called legs. The two angles that share a base of a trapezoid are called base angles.

Use Your Vocabulary 4. Cross out the figure that is NOT a trapezoid.

5. Circle the figure(s) than can be divided into two trapezoids. Then divide each figure that you circled into two trapezoids.

leg leg

base

base

base angles

trapezoid


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d.Theorems 6-19, 6-20, and 6-21

Problem 2

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167 Lesson 6-6

Finding Angle Measures in Isosceles Trapezoids

Got It? A fan has 15 angles meeting at the center. What are the measures of the base angles of the congruent isosceles trapezoids in its second ring?

Use the diagram at the right for Exercises 12–16.

12. Circle the number of isosceles triangles in each wedge. Underline the number of isosceles trapezoids in each wedge.

one two three four

13. a 5 360 4 5

14. b 5180 2

25

15. c 5 180 2 5

16. d 5 180 2 5

17. The measures of the base angles of the isosceles trapezoids are and .

Theorem 6-19 If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent.

Theorem 6-20 If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

6. If TRAP is an isosceles trapezoid with bases RA and TP,

then /T > / and /R > / .

7. Use Theorem 6-19 and your answers to Exercise 6 to draw congruence marks on the trapezoid at the right.

8. If ABCD is an isosceles trapezoid, then AC > .

9. If ABCD is an isosceles trapezoid and AB 5 5 cm, then

CD 5 cm.

10. Use Theorem 6-20 and your answer to Exercises 8 and 9 to label the diagram at the right.

Theorem 6-21 Trapezoid Midsegment Theorem If a quadrilateral is a trapezoid, then

(1) the midsegment is parallel to the bases, and

(2) the length of the midsegment is half the sum of the lengths of the bases.

11. If TRAP is a trapezoid with midsegment MN, then

(1) MN 6 6 (2) MN 5 12 Q 1 R

P A

BD

5

TP RA TP

24

15 24

78

78

78

78

102

102

102

RA


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Problem 4

Problem 3

Theorem 6-22

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Chapter 6 168

Using the Midsegment Theorem

Got It? Algebra MN is the midsegment of trapezoid PQRS. What is x? What is MN?

18. The value of x is found below. Write a reason for each step.

MN 5 12 (QR 1 PS)

2x 1 11 5 12 f10 1 (8x 2 12)g

2x 1 11 5 12 (8x 2 2)

2x 1 11 5 4x 2 1

2x 1 12 5 4x

12 5 2x

6 5 x

19. Use the value of x to find MN.

A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.

Finding Angle Measures in Kites

Got It? Quadrilateral KLMN is a kite. What are ml1, ml2, and ml3?

22. Diagonals of a kite are perpendicular, so m/1 5 .

23. nKNM > nKLM by SSS, so m/3 5 m/NKM 5 .

24. m/2 5 m/1 2 m/ by the Triangle Exterior Angle Theorem.

25. Solve for m/2.

Theorem 6-22 If a quadrilateral is a kite, then its diagonals are perpendicular.

20. If ABCD is a kite, then AC ' .

21. Use Theorem 6-22 and Exercise 20 to draw congruence marks and right angle symbol(s) on the kite at the right.

B

C

D

A

Substitute.

Trapezoid Midsegment Theorem

Simplify.

Distributive Property

Add 1 to each side.

Subtract 2x from each side.

Divide each side by 2.

MN 5 2x 1 11 5 2(6) 1 11 5 12 1 11 5 23

ml2 5 90 2 36 5 54

BD

90

36

3


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Lesson Check

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

169 Lesson 6-6


trapezoid kite base leg midsegment

Rate how well you can use properties of trapezoids and kites.

Compare and Contrast How is a kite similar to a rhombus? How is it different? Explain.

26. Place a ✓ in the box if the description fits the figure. Place an ✗ if it does not.

Kite Description Rhombus

Quadrilateral

Perpendicular diagonals

Each diagonal bisects a pair of opposite angles.

Congruent opposite sides

Two pairs of congruent consecutive sides

Two pairs of congruent opposite angles

Supplementary consecutive angles

27. How is a kite similar to a rhombus? How is it different? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________


Similar: Both are quadrilaterals with perpendicular diagonals, two pairs of

congruent consecutive sides and one pair of congruent opposite angles.

Different: Kites have no congruent opposite sides, only one pair of

congruent opposite angles, and no supplementary consecutive angles.

Answers may vary. Sample:


Quadrilaterals Graphic Organizer Kite and Trapezoid Properties, Properties of Parallelograms, Properties of Special Quads

Quadrilateral Name

Side Properties Angle Properties Diagonals Properties

Kite A quadrilateral with two distinct pairs of

congruent consecutive sides.

Trapezoid A quadrilateral with exactly one pair of

parallel sides.

Isosceles Trapezoid A trapezoid with two

congruent legs.

Parallelogram A quadrilateral with two pairs of parallel

sides.

Rhombus An equilateral parallelogram.

All of the same properties of a parallelogram


All of the same properties of a parallelogram and…

Rectangles An equiangular parallelogram.



All the same properties of a parallelogram and…

Squares An equiangular and

equilateral parallelogram. A

regular quadrilateral.



All of the same properties of a parallelogram and…

Name Class Date

6-5

Practice

Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain.

1. 2.

3. 4.

For what value of x is the figure the given special parallelogram?

5. rhombus 6. rhombus 7. rectangle

Open-Ended Given two segments with lengths x and y (x ¹ y), what special parallelograms meet the given conditions? Show each sketch.

11. One diagonal has length x, the other has length y. The diagonals intersect at right angles.

12. Both diagonals have length y and do not intersect at right angles.


43

Conditions for Rhombuses, Rectangles, and Squares

8. rhombus 9. rectangle 10. rectangle

Form G

Name Class Date

6-5

Practice (continued) Form G

For Exercises 13–16, determine whether the parallelogram is a rhombus, a rectangle, or a square. Give the most precise description in each case.

13. A parallelogram has perpendicular diagonals and angle measures of 45, 135, 45, and 135.

14. A parallelogram has perpendicular and congruent diagonals.

15. A parallelogram has perpendicular diagonals and angle measures that are all 90.

16. A parallelogram has congruent diagonals.

17. A woman is plotting out a garden bed. She measures the diagonals of the bed and finds that one is 22 ft long and the other is 23 ft long. Could the garden bed be a rectangle? Explain.

18. A man is making a square frame. How can he check to make sure the frame is square, using only a tape measure?

19. A girl cuts out rectangular pieces of cardboard for a project. She checks to see that they are rectangular by determining if the diagonals are perpendicular. Will this tell her whether a piece is a rectangle? Explain.

20. Reasoning Explain why drawing both diagonals on any rectangle will always result in two pairs of nonoverlapping congruent triangles.


44

Conditions for Rhombuses, Rectangles, and Squares

Polygons Name: A


P.2 I can use the properties of rhombuses, rectangles, and squares to solve for missing angles, sides, diagonals, etc.


1. rhombus

2. square

3. rectangle

4. rectangle

5. rectangle

6. rhombus

7. A parallelogram has perpendicular diagonals and angle measures of 45, 135, 45, and 135. What special parallelogram is it?

Polygons Name: B


P.2 I can use the properties of rhombuses, rectangles, and squares to solve for missing angles, sides, diagonals, etc.


1. rhombus

2. rhombus

3. rectangle

4. rectangle

7. rectangle

8. rectangle

7. A girl cuts out pieces of cardboard for a project. She wants to know what shape they are so she checks and determines that the diagonals are perpendicular. What special quadrilateral are the pieces?

Name Class Date

6-6

Find the measures of the numbered angles in each isosceles trapezoid.

1. 2. 3.

4. 5.

6. 7.

Algebra Find the value(s) of the variable(s) in each isosceles trapezoid.

8. 9. 10.

11. 12. 13.

Algebra Find the lengths of the segments with variable expressions.

14. 15. 16.


53

Trapezoids and Kites

Find XY in each trapezoid.

Form G Practice

Name Class Date

6-6

Practice (continued) Form G

17. is the midsegment of trapezoid WXYZ.

a. What is the value of x?

b. What is XY?

c. What is WZ?

18. Reasoning The diagonals of a quadrilateral form two acute and two obtuse angles at their intersection. Is this quadrilateral a kite? Explain.

19. Reasoning The diagonals of a quadrilateral form right angles and its side lengths are 4, 4, 6, and 6. Could this quadrilateral be a kite? Explain.

Find the measures of the numbered angles in each kite. 20. 21. 22. 23. 24. 25.

Algebra Find the value(s) of the variable(s) in each kite.

26. 27. 28.

For which value of x is each figure a Kite?

29. 30.


54

CD

Trapezoids and Kites

Special Quadrilaterals Name: _______________________________________

Identifythequadrilateralshownandthendeterminethevalueofthemissingangles.Showyourwork.

1.

2.

3.

4.

5.

Determinethevalueofxinthefollowingfigures.Thetypeofquadrilateralisgiven.6. Kite

7. Rhombus

8. Rectangle

9. Trapezoid

10. Kite

Special Quadrilaterals Name: _______________________________________

Identifythequadrilateralshownandthendeterminethevalueofthemissingangles.Showyourwork.

1.

2.

3.

4.

5.

Determinethevalueofxinthefollowingfigures.Thetypeofquadrilateralisgiven.6. Rhombus

7. Kite

8. Kite

9. Trapezoid

10. Rectangle

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