Word/Theorem Definition Example/Properties - … - 4 Square...Interior Angle Sum Theorem ......

Word/Theorem Definition Example/Properties

Quadrilateral

Square

Rectangle

Rhombus

Kite

Parallelogram

Trapezoid

Name: ________________________________________ Geometry - Quadrilaterals


Interior Angle Sum Theorem

Exterior Angle Sum Theorem


Trapezoid

Interior Angle Sum Theorem

Exterior Angle Sum Theorem

Lesson 8.6 | Quadrilateral Family 467

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8Complete the table by placing a checkmark in the appropriate row and column to associate each figure with its properties.

Quadrilateral

Trapezoid

Parallelogram

Kite

Rhom

bus

Rectangle

Square

No parallel sides

Exactly one pair of parallel sides

Two pairs of parallel sides

One pair of sides are both congruent and parallel

Two pairs of opposite sides are congruent

Exactly one pair of opposite angles are congruent

Two pairs of opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other

All sides are congruent

Diagonals are perpendicular to each other

Diagonals bisect the vertex anglesAll angles are congruent

Diagonals are congruent

PROBLEM 1 Characteristics of Quadrilaterals

Quadrilateral FamilyCategorizing Quadrilaterals

8.6

OBJECTIVESIn this lesson you will:l List the properties of quadrilaterals.l Categorize quadrilaterals based upon their properties.l Construct quadrilaterals given a diagonal.



Properties of Squares1.) All sides are ____________

2.) All angles are ____________

3.) Diagonals are ____________

4.) Opposite sides are ___________

5.) Diagonals ___________ each other.

6.) Diagonals __________ the ___________ angles.

7.) Diagonals are ____________ to each other.1

Properties ExplanationIn a Pages document, explain/prove the following properties of quadrilaterals using the given diagrams.

Make sure you label each explanation with the corresponding property.

Each explanation should be well thought out and include theorems that we have discovered this year.

This document will count as part of your test and is due on Friday, April 12th.

Squares:• Perpendicular/Parallel Line Theorem• Two pairs of parallel sides.• Diagonals are congruent.• Diagonals bisect each other.• Diagonals bisect the vertex angles. • Diagonals are perpendicular to each other.

Parallelograms:• Two pairs of opposite sides are congruent.• Two pairs of opposite angles are congruent.

Kite:• Exactly one pair of opposite angles are congruent.• Diagonals bisect the vertex angles.

Trapezoid:• Diagonals are congruent.

Interior/Exterior Angles:• Explain the Interior Angle Theorem and how it can be proven.• Explain the Exterior Angle Theorem and how it can be proven.

2

Perpendicular/Parallel Line TheoremIf two lines are perpendicular to the same line, then the two lines are parallel to each other.

418 Chapter 8 | Quadrilaterals

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GroupingOnce the students have completed Questions 1 and 2, have the groups share their methods and solutions.

2. Ramira is helping Jessica with her math homework. She tries to explain the theorem: “If two lines are perpendicular to the same line, then the two lines are parallel to each other.” Jessica doesn’t understand why this is true. Use the diagram shown and complete the proof to help Jessica understand this theorem.

!1

!2

!3

1 243

5 687

Given: !1 ! !3; !2 ! !3

Prove: !1 " !2

Statements Reasons1. !1 ! !3 1. Given

2. !2! !3 2. Given

3. "1, "2, "3, "4, "5, "6, "7, and "8 are right angles.

3. Definition of perpendicular lines

4. "1 ! "2 ! "3 ! "4 ! "5 ! "6 ! "7 ! "8 4. All right angles are congruent.

5. !1 " !25. Alternate Interior Angle

Converse Theorem

Perpendicular/Parallel Line Theorem: If two lines are perpendicular to the same line, then the two lines are parallel to each other.

3

Opposite Sides are Parallel

Lesson 8.1 | Squares and Rectangles 421

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GroupingHave the students work in groups to complete Questions 7 and 8. Once the students have completed the questions, have the groups share their methods and solutions.

7. Create a two-column proof of the statement ___

DA ! ___

CB and ___

DC ! ___

AB .

D C

BA

Given: Square ABCD Prove:

___ DA !

___ CB and

___ DC !

___ AB

Statements Reasons1. Square ABCD 1. Given

2. !D, !A, !B, and !C are right angles. 2. Definition of square

3. ___

DA " ___

AB , ___

AB " ___

BC , ___

BC " ___

CD , and

___ CD "

___ DA

3. Definition of perpendicular lines

4. ___

DA ! ___

CB and ___

DC ! ___

AB 4. Perpendicular/Parallel Line Theorem

8. If a parallelogram is a quadrilateral with opposite sides parallel, do you have enough information to conclude square ABCD is a parallelogram? Explain.

Yes. We have just proven opposite sides of a square are parallel.

Congratulations! You have just proven another property of a square!Property of a Square: Opposite sides of a square are parallel.You can now use this property as a valid reason in future proofs.

4

Proof of Congruent Diagonals in a Square


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GroupingHave the students work in pairs to complete Questions 5 and 6. Once the students have completed the questions, have the groups share their methods and solutions.

5. Create a two-column proof of the statement !DAB ! !CBA.

D C

BA

E

Given: Square ABCD with diagonals ___

AC and ___

BD intersecting at point E Prove: !DAB ! !CBA

Statements Reasons1. Square ABCD with diagonals

___ AC and

___ BD

intersecting at point E1. Given

2. "DAB and "CBA are right angles. 2. Definition of square

3. "DAB ! "CBA 3. All right angles are congruent.

4. ___

DA ! ___

CB 4. Definition of square

5. ___

AB ! ___

AB 5. Reflexive Property

6. !DAB ! !CBA 6. SAS Congruence Theorem

6. Do you have enough information to conclude ___

AC ! ___

BD ? Explain. Yes. Because

___ AC !

___ BD by CPCTC.

Congratulations! You have just proven a property of a square.Property of a Square: Diagonals of a square are congruent. You can now use this property as a valid reason in future proofs.


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GroupingHave the students work in pairs to complete Questions 5 and 6. Once the students have completed the questions, have the groups share their methods and solutions.

5. Create a two-column proof of the statement !DAB ! !CBA.

D C

BA

E


AC and ___

BD intersecting at point E Prove: !DAB ! !CBA


___ AC and

___ BD

intersecting at point E1. Given

2. "DAB and "CBA are right angles. 2. Definition of square

3. "DAB ! "CBA 3. All right angles are congruent.

4. ___

DA ! ___

CB 4. Definition of square

5. ___

AB ! ___

AB 5. Reflexive Property

6. !DAB ! !CBA 6. SAS Congruence Theorem


AC ! ___

BD ? Explain. Yes. Because

___ AC !

___ BD by CPCTC.

Congratulations! You have just proven a property of a square.Property of a Square: Diagonals of a square are congruent. You can now use this property as a valid reason in future proofs.

Prove:

5

Other Properties

Prove: Use the diagram above to conclude the diagonals of a square bisect the vertex angles.

Prove: Use the diagram above to conclude the diagonals of a square are perpendicular to each other.


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GroupingHave the students work in groups to complete Questions 9 and 10. Once the students have completed the questions, have the groups share their methods and solutions.

9. Create a two-column proof. Use !DEC and !BEA to prove ___

DE ! ___

BE and ___

CE ! ___

AE .

D C

BA

E


AC and ___

BD intersecting at point E Prove:

___ DE !

___ BE and

___ CE !

___ AE


___

AC and ___

BD intersecting at point E1. Given

2. ___

DC ! ___

AB 2. Definition of square

3. ___

DC ! ___

AB 3. Opposite sides of a square are parallel.

4. "ABD ! "CDB 4. Alternate Interior Angle Theorem

5. "CAB ! "ACD 5. Alternate Interior Angle Theorem

6. !DEC ! !BEA 6. ASA Congruence Theorem

7. ___

DE ! ___

BE 7. CPCTC

8. ___

CE ! ___

AE 8. CPCTC

10. Do you have enough information to conclude the diagonals of a square bisect each other? Explain.

Yes. The definition of bisect is to divide into two equal parts and I have just proven both segments on each diagonal are congruent.

Congratulations! You have just proven another property of a square!Property of a Square: The diagonals of a square bisect each other.You can now use this property as a valid reason in future proofs.

6

Properties of Rectangles

1.) Opposite sides are ____________ and __________

2.) All angles are __________.

3.) Diagonals are ____________


7

Practice

Chapter 8 ! Assignments 137

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Assignment Assignment for Lesson 8.1

Name _____________________________________________ Date ____________________

Squares and Rectangles Properties of Squares and Rectangles

1. In quadrilateral VWXY, segments VX and WY bisect each other, and are perpendicular and congruent. Is this enough information to conclude that quadrilateral VWXY is a square? Explain.

V

W X

Y

Z

Quadrilateral PQRS is a rectangle with diagonals PR and QS.

P

R Q

S

T

2. Name all parallel segments.

8

Practice

138 Chapter 8 ! Assignments

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8 3. Name all congruent segments.

4. Name all right angles.

5. Name all congruent angles.

6. Name all congruent triangles.


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Name _____________________________________________ Date ____________________

Squares and Rectangles Properties of Squares and Rectangles

1. In quadrilateral VWXY, segments VX and WY bisect each other, and are perpendicular and congruent. Is this enough information to conclude that quadrilateral VWXY is a square? Explain.

V

W X

Y

Z

Quadrilateral PQRS is a rectangle with diagonals PR and QS.

P

R Q

S

T

2. Name all parallel segments.

9

602 Chapter 8 ! Skills Practice

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8 5. Given: n ! r and r ! q 6. Given: b ! x and k ! b

y n q v

r

b

x

e

z

k

Complete each statement for square GKJH.

E

G K

H J

7. ___

GK ! ___

KJ ! ___

JH ! ____

HG

8. "KGH ! " ! " ! " ! " ! " ! " ! "

9. "GEK, " , " , " , " , " , " , and " are right angles.

10. ___

GK " and ____

GH "

11. ___

GE ! ! !

12. " ! " ! " ! " ! " ! " ! " ! "

Practice

10

Chapter 8 ! Skills Practice 603

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Name _____________________________________________ Date ____________________

Complete each statement for rectangle TMNU.

T

M N

U

L

13. ____

MN ! ___

TU and ___

MT ! ___

NU

14. !NMT ! ! ! ! ! !

15. !MTU, ! , ! , and ! are right angles.

16. ____

MN " and ___

MT "

17. ____

MU !

18. ___

ML ! ! !

Construct each quadrilateral using the given information.

19. Use ___

AB to construct square ABCD with diagonals ___

AC and ___

BD intersecting at point E.

A B

D C

A B

E

Practice

11

Find the measures of the missing sides/diagonals.

A

C

B

D

E

AB = 8 inAD = 6 in

AC =

AE =

DB =

DE =

Practice

12

Properties of Parallelograms

1.) Opposite sides are ____________ and _____________

2.) Opposite angles of a parallelogram are ___________


13

Opposite Sides & Angles of a Parallelogram Congruency Proof


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Use only a compass and a straightedge to create the geometric figure.

GroupingHave the students work in groups to complete Questions 2 through 4.

2. Use ___

PA to construct parallelogram PARG with diagonals ___

PR and ___

AG intersecting at point M.

AP

AP

G R

M

3. To prove opposite sides of a parallelogram are congruent, which triangles would you prove congruent?

P

G

M

R

A

I can prove either !PGR ! !RAP or ! APG ! !GRA.

Lesson 8.2 | Parallelograms and Rhombi 431

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GroupingOnce the students have completed Questions 2 through 4, have the groups share their methods and solutions.


8

4. Use !PGR and !RAP in the parallelogram from Question 3 to prove opposite sides of a parallelogram are congruent. Create a two-column proof of the statement

___ PG !

___ AR and

___ GR !

___ PA .

Given: Parallelogram PARG with diagonals ___

PR and ___

AG intersecting at point M Prove:

___ PG !

___ AR and

___ GR !

___ PA

Statements Reasons1. Parallelogram PARG with diagonals

___

PR and ___

AG intersecting at point M1. Given

2. ___

PG ! ___

AR and ___

GR ! ___

PA 2. Definition of parallelogram

3. "GPR " "ARP and "APR " "GRP 3. Alternate Interior Angle Theorem

4. ___

PR " ___

PR 4. Reflexive Property

5. !PGR " !RAP 5. ASA Congruence Theorem

6. ___

PG " ___

RA and ___

GR " ___

AP 6. CPCTC

Congratulations! You have just proven a property of a parallelogram!Property of a Parallelogram: Opposite sides of a parallelogram are congruent. You can now use this property as a valid reason in future proofs.

5. Do you have enough information to conclude "PGR ! "RAP ? Explain. Yes. There is enough information because "PGR " "RAP by CPCTC.

6. What additional angles would you need to show congruent to prove opposite angles of a parallelogram are congruent? What two triangles do you need to prove congruent?

I would also need to show "GPA " "ARG. I can prove these angles congruent by CPCTC if I can prove ! APG " !GRA.


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7. Use ! APG and !GRA in the diagram from Question 3 to prove opposite angles of a parallelogram are congruent. Create a two-column proof of the statement "GPA ! "ARG.


PR and ___

AG intersecting at point M Prove: "GPA ! "ARG (You have already proven "PGR ! "RAP in Question 5.)


___

PR and ___


2. ___

PG ! ___

AR and ___

PA ! ___

GR 2. Definition of parallelogram

3. "PAG " "RGA and "PGA " "RAG 3. Alternate Interior Angle Theorem

4. ___

GA " ___

GA 4. Reflexive Property of "

5. !APG " !GRA 5. ASA Congruence Theorem

6. "GPA " "ARG 6. CPCTC

Congratulations! You have just proven another property of a parallelogram!Property of a Parallelogram: Opposite angles of a parallelogram are congruent.You can now use this property as a valid reason in future proofs.

8. Write a paragraph proof to conclude the diagonals of a parallelogram bisect each other. Use the parallelogram in Question 3.

I can prove !PMA " !RMG by the AAS Congruence Theorem, so ____

PM " ____

RM and ____

GM " ____

AM by CPCTC, proving the diagonals of a parallelogram bisect each other.

Congratulations! You have just proven another property of a parallelogram!Property of a Parallelogram: The diagonals of a parallelogram bisect each other. You can now use this property as a valid reason in future proofs.

9. Ray told his math teacher that he thinks a quadrilateral is a parallelogram if only one pair of opposite sides is known to be both congruent and parallel. Is Ray correct? Write a paragraph proof to verify Ray’s conjecture. Use the diagram from Question 3.

Ray is correct.

He can prove !PAR ! !RGP by the ASA Congruence Theorem, so ___

PG ! ___

AR and "PRA ! "RPG by CPCTC. Then, he can use the Alternate Interior Angle Converse Theorem to show

___ PG #

___ AR , proving quadrilateral PARG is a parallelogram.


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7. Use ! APG and !GRA in the diagram from Question 3 to prove opposite angles of a parallelogram are congruent. Create a two-column proof of the statement "GPA ! "ARG.


PR and ___

AG intersecting at point M Prove: "GPA ! "ARG (You have already proven "PGR ! "RAP in Question 5.)


___

PR and ___


2. ___

PG ! ___

AR and ___

PA ! ___

GR 2. Definition of parallelogram

3. "PAG " "RGA and "PGA " "RAG 3. Alternate Interior Angle Theorem

4. ___

GA " ___

GA 4. Reflexive Property of "

5. !APG " !GRA 5. ASA Congruence Theorem

6. "GPA " "ARG 6. CPCTC

Congratulations! You have just proven another property of a parallelogram!Property of a Parallelogram: Opposite angles of a parallelogram are congruent.You can now use this property as a valid reason in future proofs.

8. Write a paragraph proof to conclude the diagonals of a parallelogram bisect each other. Use the parallelogram in Question 3.

I can prove !PMA " !RMG by the AAS Congruence Theorem, so ____

PM " ____

RM and ____

GM " ____

AM by CPCTC, proving the diagonals of a parallelogram bisect each other.

Congratulations! You have just proven another property of a parallelogram!Property of a Parallelogram: The diagonals of a parallelogram bisect each other. You can now use this property as a valid reason in future proofs.

9. Ray told his math teacher that he thinks a quadrilateral is a parallelogram if only one pair of opposite sides is known to be both congruent and parallel. Is Ray correct? Write a paragraph proof to verify Ray’s conjecture. Use the diagram from Question 3.

Ray is correct.

He can prove !PAR ! !RGP by the ASA Congruence Theorem, so ___

PG ! ___

AR and "PRA ! "RPG by CPCTC. Then, he can use the Alternate Interior Angle Converse Theorem to show

___ PG #

___ AR , proving quadrilateral PARG is a parallelogram.

14

Properties of Kites1.) Has two sets of ______________ ___________ sides.

2.) _______________ angles are ____________.

3.) ____________ angles are ________________.

4.) Diagonals are ________________.

5.) The diagonal from the vertex angles ___________ the other diagonal.

6.) Diagonals from the __________ angles __________ the ___________ angles.

15

Proof :Non-Vertex Angles of a Kite are Congruent.

438 Chapter 8 | Quadrilaterals©

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Once the students have completed Questions 2 through 4, have the groups share their methods and solutions.

2. Construct kite KITE with diagonals __

IE and ___

KT intersecting at point S.

I

K

S T

E

3. To prove one pair of opposite angles of a kite is congruent, which triangles in the kite would you prove congruent? Explain your reasoning.

I

E

SK T

I can prove !KIT ! !KET to show "KIT ! "KET by CPCTC.

4. Prove one pair of opposite angles of a kite congruent. Given: Kite KITE with diagonals

___ KT and

__ IE intersecting at point S.

Prove: "KIT ! "KET

Statements Reasons1. Kite KITE with diagonals

___ KT and

__ IE

intersecting at point S1. Given

2. __

KI ! ___

KE 2. Definition of kite

3. __

TI ! ___

TE 3. Definition of kite

4. ___

KT ! ___

KT 4. Reflexive Property

5. !KIT ! !KET 5. SSS Congruence Theorem

6. "KIT ! "KET 6. CPCTC

438 Chapter 8 | Quadrilaterals©

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Once the students have completed Questions 2 through 4, have the groups share their methods and solutions.

2. Construct kite KITE with diagonals __

IE and ___

KT intersecting at point S.

I

K

S T

E

3. To prove one pair of opposite angles of a kite is congruent, which triangles in the kite would you prove congruent? Explain your reasoning.

I

E

SK T

I can prove !KIT ! !KET to show "KIT ! "KET by CPCTC.

4. Prove one pair of opposite angles of a kite congruent. Given: Kite KITE with diagonals

___ KT and

__ IE intersecting at point S.

Prove: "KIT ! "KET

Statements Reasons1. Kite KITE with diagonals

___ KT and

__ IE

intersecting at point S1. Given

2. __

KI ! ___

KE 2. Definition of kite

3. __

TI ! ___

TE 3. Definition of kite

4. ___

KT ! ___

KT 4. Reflexive Property

5. !KIT ! !KET 5. SSS Congruence Theorem

6. "KIT ! "KET 6. CPCTC

Proof: Diagonal Connecting Vertex Angles Bisects the Vertex Angles.

Lesson 8.3 | Kites and Trapezoids 439

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GroupingHave the students work in groups to complete Questions 5through 9. Once the students have completed the questions, have the groups share their methods and solutions.

NoteThe last question, Question 9, asks students to revisit Question 1, where they listed the known properties of a kite, to verify all properties proven in this problem are now listed.

8

Congratulations! You have just proven a property of a kite!Property of a Kite: One pair of opposite angles is congruent.You are now able to use this property as a valid reason in future proofs.


KT bisects !IKE and !ITE? Explain your reasoning.

Yes. There is enough information to conclude because !IKT ! !EKT and !ITK ! !ETK by CPCTC.

6. What two triangles could you use to prove __

IS ! ___

ES ? I can first prove "KIS ! "KES, or "ITS ! "ETS by the SAS Congruence Theorem, and

then __

IS ! ___

ES by CPCTC.

7. If __

IS ! ___

ES , is that enough information to determine that one diagonal of a kite bisects the other diagonal? Explain.

Yes. If __

IS ! ___

ES , then by the definition of bisect, diagonal ___

KT bisects diagonal __

IE .

8. Write a paragraph proof to conclude the diagonals of a kite are perpendicular to each other.

I can first prove "KIS ! "KES by the SAS Congruence Theorem, and then !KIS ! !KES by CPCTC. These angles also form a linear pair by the Linear Pair Postulate. The angles are supplementary by the definition of a linear pair, and two angles that are both congruent and supplementary are right angles. If they are right angles, then the lines forming the angles must be perpendicular.

Congratulations! You have just proven another property of a kite!Property of a Kite: The diagonals of a kite are perpendicular to each other. You are now able to use this property as a reason in future proofs.

9. Revisit Question 1 to make sure you have listed all of the properties of a kite.

Prove:

16

Parts of a Trapezoid1.) Bases:

2.) Legs:

3.) Base Angles:

4.) Isosceles Trapezoid:

17

Properties of Isosceles Trapezoids1.) Legs are _____________.

2.) Bases are ______________.

3.) ____________ angles are ________________.

4.) Diagonals are ________________.

18

Proof of Congruent Diagonals in an Isosceles Trapezoid (Given Base Angles are Congruent)

Lesson 8.3 | Kites and Trapezoids 443

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5. Use the trapezoid shown to prove each statement.

T

R

P

A

Given: Isosceles Trapezoid TRAP with ___

TP ! ___

RA , ___

TR " ___

PA , and diagonals

___ TA and

___ PR .

Prove: ___

TA " ___

PR

Statements Reasons1. Isosceles Trapezoid TRAP with

___ TP !

___ RA ,

___

TR # ___

PA , and diagonals ___

TA and ___

PR 1. Given

2. !RTP # !APT 2. Base angles of an isosceles trapezoid are congruent.

3. ___

TP # ___

TP 3. Reflexive Property

4. "RTP # "APT 4. SAS Congruence Theorem

5. ___

TA # ___

PR 5. CPCTC

Given: Trapezoid TRAP with ___

TP ! ___

RA , and diagonals ___

TA " ___

PR Prove: Trapezoid TRAP is isosceles

To prove the converse, auxiliary lines must be drawn such that ___

RA is extended to intersect a perpendicular line passing through point T perpendicular to ___

RA ( ___

TE ) and intersect a second perpendicular line passing through point P perpendicular to

___ RA (

___ PZ ).

T E

R

P Z

A

Notice that quadrilateral TEZP is a rectangle.

19

Practice


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Skills Practice Skills Practice for Lesson 8.3

Name _____________________________________________ Date ____________________

Kites and Trapezoids Properties of Kites and Trapezoids

VocabularyWrite the term from the box that best completes each statement.

base angles of a trapezoid biconditional statement isosceles trapezoid

1. The are either pair of angles of a trapezoid that share a base as a common side.

2. A(n) is a trapezoid with congruent non-parallel sides.

3. A(n) is a statement that contains if and only if.

Problem SetComplete each statement for kite PRSQ.

1. ___

PQ ! ___

QS and ___

PR ! ___

SR

QR

S

T

P

2. !QPR ! !

3. ___

PT !

4. !PQT ! ! and !PRT ! !

624 Chapter 8 ! Skills Practice

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8Complete each statement for trapezoid UVWX.

5. The bases are ___

UV and ____

WX .

X

U V

W

6. The pairs of base angles are ! and ! , and ! and ! .

7. The legs are and .

8. The vertices are , , , and .

Construct each quadrilateral using the given information.

9. Construct kite QRST with diagonals ___

QS and ___

RT intersecting at point M.

Q S

R T

Q

T M

S

R

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Name _____________________________________________ Date ____________________

Use the given figure to answer each question.

15. The figure shown is a kite with !DAB ! !DCB. Which sides of the kite are congruent?

A

B

D C

___

AB and ___

CB are congruent.

___

AD and ___

CD are congruent.

16. The figure shown is a kite with ___

FG ! ___

FE . Which of the kite’s angles are congruent?

E

F

G H

17. Given that IJLK is a kite, what kind of triangles are formed by diagonal __

IL ?

I J

K

L


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Name _____________________________________________ Date ____________________

Use the given figure to answer each question.

21. The figure shown is an isosceles trapezoid with ___

AB || ___

CD . Which sides are congruent?

A B

C D

___

AC and ___

BD are congruent.

22. The figure shown is an isosceles trapezoid with ___

EH ! ___

FG . Which sides are parallel?

E F

H G

23. The figure shown is an isosceles trapezoid with __

IJ ! ___

KL . What are the bases?

I

J

K

L

24. The figure shown is an isosceles trapezoid with ____

MP ! ___

NO . What are the pairs of base angles?

PM

NO

21


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8Quadrilateral WXYZ is an isosceles trapezoid.

YX

ZW

V

5. If m!XWZ ! 66°, what is m!YZW? Explain.

6. If the length of ____

WY is 10 inches, what is ZX? Explain.

7. If the length of ____

WX is 7 inches, what is ZY? Explain.


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Name _____________________________________________ Date ____________________ Kites and Trapezoids Properties of Kites and Trapezoids

Quadrilateral ABCD is a kite.

A

B

C

D

E

1. If m!ABC ! 95°, what is m!ADC? Explain.

2. If m!BCE ! 34°, what is m!EBC? Explain.

3. If the length of ___

AB is 16 feet, what is AD? Explain.

4. If the length of ___

BD is 25 feet, what is ED? Explain.

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8

Carson drew a quadrilateral and added one diagonal as shown. He concluded that the sum of the measures of the interior angles of a quadrilateral must be equal to 360º.

1. Describe Carson’s reasoning.

Juno drew a quadrilateral and added two diagonals as shown. She concluded that the sum of the measures of the interior angles of a quadrilateral must be equal to 720º.

2. Describe Juno’s reasoning.

3. Who is correct? Explain.

Decomposing Polygons: Interior and Exterior Angles

23

Interior Angle SumFormula:

Some Problems (solve for x):

20x + 15

3x + 90

4x + 101

27x + 50

37x

30x - 20 116˚

103˚ 172˚

141˚153˚

147˚

x˚

24

Working Backwards

If the sum of the angles in a polygon is 1620˚, how many sides does it have?



25

Each Interior Angle in a Regular PolygonFormula:

Find each interior angle in each polygon:Regular Decagon:

Regular Octagon:

Regular Hexagon:

Regular Quadrilateral:

26

Working BackwardsIf each interior angle in a regular polygon is 150˚, how many sides does it have?

If each interior angle in a regular polygon is 144˚, how many sides does it have?

If each interior angle in a regular polygon is 135˚, how many sides does it have?

27

Triangle:Calculate the sum of the exterior angle measures of a triangle by completing each step.Step 1: Draw a triangle and extend each side to locate an exterior angle at each vertex.

Step 2: Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a triangle.

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Quadrilateral

Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 461

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2. Calculate the sum of the exterior angle measures of a quadrilateral by completing each step.

Step 1: Draw a quadrilateral and extend each side to locate an exterior angle at each vertex.

Step 2: Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a quadrilateral.

3. Calculate the sum of the exterior angle measures of a pentagon by completing each step.

Step 1: Draw a pentagon and extend each side to locate an exterior angle at each vertex.

29

PentagonCalculate the sum of the exterior angle measures of a pentagon by completing each step.Step 1: Draw a pentagon and extend each side to locate an exterior angle at each vertex.

Step 2: Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a pentagon.

30

HexagonCalculate the sum of the exterior angle measures of a hexagon by completing each step.Step 1: Draw a hexagon and extend each side to locate an exterior angle at each vertex.

Step 2: Use the formula for the sum of interior angle measures of a polygon and the Linear Pair Postulate to calculate the sum of the exterior angle measures of a hexagon.

31

Putting it Together


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5. Complete the table.

Number of sides of the polygon

3 4 5 6 7 15

Number of linear pairs formed

Sum of measures of linear pairs

Sum of measures of interior angles

Sum of measures of exterior angles

6. When you calculated the sum of the exterior angle measures in the 15-sided polygon, did you need to know anything about the number of linear pairs, the sum of the linear pair measures, or the sum of the interior angle measures of the 15-sided polygon? Explain.

7. If a polygon has 100 sides, calculate the sum of the exterior angle measures. Explain how you calculated your answer.

8. What is the sum of the exterior angle measures of an n-sided polygon?

9. If the sum of the exterior angle measures of a polygon is 360!, how many sides does the polygon have? Explain how you got this answer.

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Number of sides of the polygon

3 4 5 6 7 15

Number of linear pairs formed

Sum of measures of linear pairs



6. When you calculated the sum of the exterior angle measures in the 15-sided polygon, did you need to know anything about the number of linear pairs, the sum of the linear pair measures, or the sum of the interior angle measures of the 15-sided polygon? Explain.

7. If a polygon has 100 sides, calculate the sum of the exterior angle measures. Explain how you calculated your answer.

8. What is the sum of the exterior angle measures of an n-sided polygon?

9. If the sum of the exterior angle measures of a polygon is 360!, how many sides does the polygon have? Explain how you got this answer.

33

Regular Polygons


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10. Explain why the sum of the exterior angle measures of any polygon is always equal to 360!.

PROBLEM 2 Regular Polygons 1. Calculate the measure of each exterior angle of an equilateral triangle.

Explain your reasoning.

2. Calculate the measure of each exterior angle of a square. Explain your reasoning.

3. Calculate the measure of each exterior angle of a regular pentagon. Explain your reasoning.

4. Calculate the measure of each exterior angle of a regular hexagon. Explain your reasoning.

5. Complete the table shown to look for a pattern.

Number of sides of a regular polygon

3 4 5 6 7 15


Measure of each interior angle

Measure of each exterior angle

34

Regular Polygons


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10. Explain why the sum of the exterior angle measures of any polygon is always equal to 360!.

PROBLEM 2 Regular Polygons 1. Calculate the measure of each exterior angle of an equilateral triangle.

Explain your reasoning.

2. Calculate the measure of each exterior angle of a square. Explain your reasoning.

3. Calculate the measure of each exterior angle of a regular pentagon. Explain your reasoning.

4. Calculate the measure of each exterior angle of a regular hexagon. Explain your reasoning.

5. Complete the table shown to look for a pattern.

Number of sides of a regular polygon

3 4 5 6 7 15



Measure of each exterior angle


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6. When you calculated the measure of each exterior angle in the 15-sided regular polygon, did you need to know anything about the measure of each interior angle? Explain.

7. If a regular polygon has 100 sides, calculate the measure of each exterior angle. Explain how you calculated your answer.

8. What is the measure of each exterior angle of an n-sided regular polygon?

9. If the measure of each exterior angle of a regular polygon is 18!, how many sides does the polygon have? Explain how you calculated your answer.

35

Practice


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Name _____________________________________________ Date ____________________

Exterior and Interior Angle Measurement Interactions Sum of the Exterior Angle Measures of a Polygon

Use the figure below to answer each question.

1

2

34

5

6

1. What is the sum of the measures of angles 1 and 4? Explain your reasoning.



4. What is the sum of the measures of angles 1, 2, 3, 4, 5, and 6? Explain your reasoning.

36

Practice


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8 5. What is the sum of the measures of angles 1, 2, and 3? Explain your reasoning.

6. What is the difference of the sum of the measures of angles 1, 2, 3, 4, 5, and 6 and the sum of the measures of angles 1, 2, and 3? What does this demonstrate?

7. If a regular polygon has 30 sides, what is the measure of each exterior angle? Explain your reasoning.

8. The degree measure of each exterior angle of a regular octagon is represented by the expression 7x ! 4. Solve for x.

37

As always, you must start with what you know to be true. The Triangle Sum Theorem states that the sum of the three interior angles of any triangle is equal to 180. You can use this information to calculate the sum of the interior angles of other polygons.Calculate the sum of the interior angle measures of a quadrilateral by completing each step.

Step 1: Draw a quadrilateral.

Step 2: Draw all possible diagonals using only one vertex of the quadrilateral. Remember, a diagonal is a line segment connecting non-adjacent vertices.

Step 3: How many triangles are formed when the diagonal(s) divide the quadrilateral?

Step 4: If the sum of the interior angle measures of each triangle is 180, what is the sum of all the interior angle measures of the triangles formed by the diagonal(s)?

38

Calculate the sum of the interior angle measures of a pentagon by completing each step.Step 1: Draw a pentagon.

Step 2: Draw all possible diagonals using only one vertex of the pentagon.

Step 3: How many triangles are formed when the diagonal(s) divide the pentagon?


39

Calculate the sum of the interior angle measures of a hexagon by completing each step.Step 1: Draw a hexagon.

Step 2: Draw all possible diagonals using one vertex of the hexagon.

Step 3: How many triangles are formed when the diagonal(s) divide the hexagon?


40

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Lesson 8.4 | Decomposing Polygons 455

8

Step 2: Draw all possible diagonals using one vertex of the hexagon.

Step 3: How many triangles are formed when the diagonal(s) divide the hexagon?

Step 4: If the sum of the interior angle measures of each triangle is 180º, what is the sum of all the interior angle measures of the triangles formed by the diagonal(s)?

4. Complete the table shown.

Number of sides of the polygon 3 4 5 6

Number of diagonals drawn

Number of triangles formed

Sum of the measures of the interior angles

5. What pattern do you notice between the number of possible diagonals drawn from one vertex of the polygon, and the number of triangles formed by those diagonals?

6. Compare the number of sides of the polygon to the number of possible diagonals drawn from one vertex. What do you notice?

7. Compare the number of sides of the polygon to the number of triangles formed by drawing all possible diagonals from one vertex. What do you notice?

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8

8. What pattern do you notice about the sum of the interior angle measures of a polygon as the number of sides of each polygon increases by 1?

9. Predict the number of possible diagonals drawn from one vertex and the number of triangles formed for a seven-sided polygon using the table you completed.

10. Predict the sum of all the interior angle measures of a seven-sided polygon using the table your completed.

11. Continue the pattern to complete the table.





12. When you calculated the number of triangles formed in the 16-sided polygon, did you need to know how many triangles were formed in a 15-sided polygon first? Explain your reasoning.

13. If a polygon has 100 sides, how many triangles are formed by drawing all possible diagonals from one vertex? Explain.

14. What is the sum of all the interior angle measures of a 100-sided polygon? Explain your reasoning.

15. If a polygon has n sides, how many triangles are formed by drawing all diagonals from one vertex? Explain.

16. What is the sum of all the interior angle measures of an n-sided polygon? Explain your reasoning.

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8. What pattern do you notice about the sum of the interior angle measures of a polygon as the number of sides of each polygon increases by 1?

9. Predict the number of possible diagonals drawn from one vertex and the number of triangles formed for a seven-sided polygon using the table you completed.

10. Predict the sum of all the interior angle measures of a seven-sided polygon using the table your completed.

11. Continue the pattern to complete the table.





12. When you calculated the number of triangles formed in the 16-sided polygon, did you need to know how many triangles were formed in a 15-sided polygon first? Explain your reasoning.

13. If a polygon has 100 sides, how many triangles are formed by drawing all possible diagonals from one vertex? Explain.

14. What is the sum of all the interior angle measures of a 100-sided polygon? Explain your reasoning.

15. If a polygon has n sides, how many triangles are formed by drawing all diagonals from one vertex? Explain.

16. What is the sum of all the interior angle measures of an n-sided polygon? Explain your reasoning.

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Lesson 8.4 | Decomposing Polygons 457

8

17. Use the formula to calculate the sum of all the interior angle measures of a polygon with 32 sides.

18. If the sum of all the interior angle measures of a polygon is 9540º, how many sides does the polygon have? Explain your reasoning.

1. Use the formula developed in Problem 2, Question 16 to calculate the sum of the all the interior angle measures of a decagon.

2. Calculate each interior angle measure of a decagon if each interior angle is congruent. How did you calculate your answer?


Number of sides of regular polygon 3 4 5 6 7 8



4. If a regular polygon has n sides, write a formula to calculate the measure of each interior angle.

5. Use the formula to calculate each interior angle measure of a regular 100-sided polygon.

PROBLEM 3 Sum of the Interior Angle Measures of a Regular Polygon

43

Practice


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Name _____________________________________________ Date ____________________ Decomposing Polygons Sum of the Interior Angle Measures of a Polygon

Determine the measure of an interior angle of the given regular polygon.

1. regular nonagon 2. regular decagon

3. regular 15-gon 4. regular 47-gon

Determine the measure of the missing angle in each figure.

5.

108°

166°

135°

90°

121°

?

6. 135°

?

146°

142°

113°

161°99°

128°


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Name _____________________________________________ Date ____________________ Decomposing Polygons Sum of the Interior Angle Measures of a Polygon

Determine the measure of an interior angle of the given regular polygon.

1. regular nonagon 2. regular decagon

3. regular 15-gon 4. regular 47-gon

Determine the measure of the missing angle in each figure.

5.

108°

166°

135°

90°

121°

?

6. 135°

?

146°

142°

113°

161°99°

128°

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8 7. Use the figure to answer each question.

7x – 12

5x

4x + 2

2x

4x

P

QR

S

T

a. What is the sum of the measures of the interior angles of the polygon?

b. What is the value of x?

c. What is the measure of !PTS?

d. What is the measure of angle !RQP?

45

Review


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8Complete the table by placing a checkmark in the appropriate row and column to associate each figure with its properties.

Quadrilateral

Trapezoid

Parallelogram

Kite

Rhom

bus

Rectangle

Square

No parallel sides

Exactly one pair of parallel sides

Two pairs of parallel sides

One pair of sides are both congruent and parallel

Two pairs of opposite sides are congruent

Exactly one pair of opposite angles are congruent

Two pairs of opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other

All sides are congruent

Diagonals are perpendicular to each other

Diagonals bisect the vertex anglesAll angles are congruent

Diagonals are congruent

PROBLEM 1 Characteristics of Quadrilaterals

Quadrilateral FamilyCategorizing Quadrilaterals

8.6

OBJECTIVESIn this lesson you will:l List the properties of quadrilaterals.l Categorize quadrilaterals based upon their properties.l Construct quadrilaterals given a diagonal.

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True or False


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Determine whether each statement is true or false. If it is false, explain why.

1. A square is also a rectangle.

2. A rectangle is also a square.

3. The base angles of a trapezoid are congruent.

4. A parallelogram is also a trapezoid.

5. A square is a rectangle with all sides congruent.

6. The diagonals of a trapezoid are congruent.

7. A kite is also a parallelogram.

8. The diagonals of a rhombus bisect each other.

PROBLEM 3 True or False

47

True or False


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1. A square is also a rectangle.

2. A rectangle is also a square.

3. The base angles of a trapezoid are congruent.

4. A parallelogram is also a trapezoid.

5. A square is a rectangle with all sides congruent.

6. The diagonals of a trapezoid are congruent.

7. A kite is also a parallelogram.

8. The diagonals of a rhombus bisect each other.

PROBLEM 3 True or False

48

True or False

9. If the diagonals of a quadrilateral are congruent, then the quadrilateral is a square.

10. If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram.

11. Suppose a quadrilateral has two pairs of congruent sides. Must the quadrilateral be a parallelogram? Explain. Make a sketch.

49

Interior Exterior Angle Measure

1204 Chapter 8 ! Assessments

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End of Chapter Test PAGE 2

3. In some quadrilaterals the diagonals are perpendicular to each other. For which quadrilaterals is this true?

4. Draw a Venn diagram describing the relationship between all of the quadrilaterals. Show the relationship between the parallelograms.

5. Chelsea drew a 16-sided polygon.

a. Calculate the sum of the interior angles of the figure.

b. Suppose the figure is a regular polygon. Use the formula to calculate each interior angle measure.

c. What is the sum of the exterior angles of the figure?

d. If the figure is a regular polygon, what is the measure of each exterior angle?

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More Interior Exterior Angles

Chapter 8 ! Assessments 1205

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End of Chapter Test PAGE 3

Name ________________________________________________________ Date __________________________

6. Each measure of an interior angle of a regular polygon is 140°. Determine the number of sides of the polygon. Show your work.

7. The measure of each exterior angle of a regular polygon is 24°. Determine the number of sides of the polygon. Show your work.

8. Determine the measure of an exterior angle of a regular octagon. Show your work.


9. If the diagonals of a quadrilateral are congruent, then the quadrilateral is a square.

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1210 Chapter 8 ! Assessments

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Standardized Test Practice PAGE 4

16. Consider the biconditional statement: A rectangle is a square if and only if the four sides are congruent.

What is the converse of the original statement?

a. If the rectangle is a square, then the four sides are not congruent.

b. If the four sides of a square are congruent, then the rectangle is a square.

c. If the square has four congruent sides, then the square is a rectangle.

d. If there are four congruent sides, then the square is a rectangle.

17. What shape is both a rhombus and a rectangle?

a. a square

b. a kite

c. a trapezoid

d. a triangle

18. In the figure shown, ABCD is a parallelogram, AC ! 16 inches, and BD ! 30 inches. Determine BC.

B C

A D

a. 31 inches

b. 21 inches

c. 18 inches

d. Cannot be determined.

19. A regular polygon has exterior angles that measure approximately 51.43˚. How many sides does the polygon have?

a. 3

b. 5

c. 6

d. 7

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Chapter 8 ! Assessments 1211

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Standardized Test Practice PAGE 5

Name ________________________________________________________ Date __________________________

20. What is the perimeter of the kite shown?

10 cm 3 cm

4 cm

4 cm

W Y

X

Z

a. 10.00 cm

b. 21.54 cm

c. 31.54 cm

d. 63.08 cm

53

Word/Theorem Definition Example/Properties - … - 4 Square...Interior Angle Sum Theorem ......

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