Lesson 3: (3.4) Parallel Lines and the Triangle Angle-Sum Theorem.
Word/Theorem Definition Example/Properties - … - 4 Square...Interior Angle Sum Theorem ......
Transcript of 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
Word/Theorem Definition Example/Properties
Interior Angle Sum Theorem
Exterior Angle Sum Theorem
Word/Theorem Definition Example/Properties
Trapezoid
Interior Angle Sum Theorem
Exterior Angle Sum Theorem
Lesson 8.6 | Quadrilateral Family 467
© 2
010
Car
negi
e Le
arni
ng, I
nc.
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.
Name: ________________________________________ Geometry - Quadrilaterals
Name: ________________________________________ Geometry - Quadrilaterals
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
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
420 Chapter 8 | Quadrilaterals
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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.
420 Chapter 8 | Quadrilaterals
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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.
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.
422 Chapter 8 | Quadrilaterals
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
Given: Square ABCD with diagonals ___
AC and ___
BD intersecting at point E Prove:
___ DE !
___ BE and
___ CE !
___ AE
Statements Reasons1. Square ABCD with diagonals
___
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 ____________
4.) Diagonals ___________ each other.
7
Practice
Chapter 8 ! Assignments 137
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8 3. Name all congruent segments.
4. Name all right angles.
5. Name all congruent angles.
6. Name all congruent triangles.
Chapter 8 ! Assignments 137
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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.
9
602 Chapter 8 ! Skills Practice
© 2
010
Car
negi
e Le
arni
ng, I
nc.
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
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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 ___________
3.) Diagonals ___________ each other.
13
Opposite Sides & Angles of a Parallelogram Congruency Proof
430 Chapter 8 | Quadrilaterals
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
GroupingOnce the students have completed Questions 2 through 4, have the groups share their methods and solutions.
GroupingHave the students work in groups to complete Questions 5 through 10.
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.
432 Chapter 8 | Quadrilaterals
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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.
Given: Parallelogram PARG with diagonals ___
PR and ___
AG intersecting at point M Prove: "GPA ! "ARG (You have already proven "PGR ! "RAP in Question 5.)
Statements Reasons1. Parallelogram PARG with diagonals
___
PR and ___
AG intersecting at point M1. Given
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.
432 Chapter 8 | Quadrilaterals
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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.
Given: Parallelogram PARG with diagonals ___
PR and ___
AG intersecting at point M Prove: "GPA ! "ARG (You have already proven "PGR ! "RAP in Question 5.)
Statements Reasons1. Parallelogram PARG with diagonals
___
PR and ___
AG intersecting at point M1. Given
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©
201
0 C
arne
gie
Lear
ning
, Inc
.
8
Use only a compass and a straightedge to create the geometric figure.
GroupingHave the students work in groups to complete Questions 2 through 4.
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©
201
0 C
arne
gie
Lear
ning
, Inc
.
8
Use only a compass and a straightedge to create the geometric figure.
GroupingHave the students work in groups to complete Questions 2 through 4.
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
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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.
5. Do you have enough information to conclude ___
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
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
Chapter 8 ! Skills Practice 623
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
© 2
010
Car
negi
e Le
arni
ng, I
nc.
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
20
Chapter 8 ! Skills Practice 627
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
Chapter 8 ! Skills Practice 629
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
142 Chapter 8 ! Assignments
© 2
010
Car
negi
e Le
arni
ng, I
nc.
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.
Chapter 8 ! Assignments 141
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
Assignment Assignment for Lesson 8.3
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.
22
© 2
010
Car
negi
e Le
arni
ng, I
nc.
452 Chapter 8 | Quadrilaterals
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?
If the sum of the angles in a polygon is 1260˚, how many sides does it have?
If the sum of the angles in a polygon is 540˚, 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.
28
Quadrilateral
Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 461
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 463
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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.
32
Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 463
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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.
33
Regular Polygons
464 Chapter 8 | Quadrilaterals
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
Sum of measures of exterior angles
Measure of each interior angle
Measure of each exterior angle
34
Regular Polygons
464 Chapter 8 | Quadrilaterals
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
Sum of measures of exterior angles
Measure of each interior angle
Measure of each exterior angle
Lesson 8.5 | Exterior and Interior Angle Measurement Interactions 465
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
Chapter 8 ! Assignments 147
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
Assignment Assignment for Lesson 8.5
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.
2. What is the sum of the measures of angles 2 and 5? Explain your reasoning.
3. What is the sum of the measures of angles 3 and 6? 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
148 Chapter 8 ! Assignments
© 2
010
Car
negi
e Le
arni
ng, I
nc.
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?
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)?
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?
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)?
40
© 2
010
Car
negi
e Le
arni
ng, I
nc.
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?
41
© 2
010
Car
negi
e Le
arni
ng, I
nc.
456 Chapter 8 | Quadrilaterals
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.
Number of sides of the polygon 7 8 9 16
Number of diagonals drawn
Number of triangles formed
Sum of the measures of the interior angles
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.
42
© 2
010
Car
negi
e Le
arni
ng, I
nc.
456 Chapter 8 | Quadrilaterals
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.
Number of sides of the polygon 7 8 9 16
Number of diagonals drawn
Number of triangles formed
Sum of the measures of the interior angles
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.
© 2
010
Car
negi
e Le
arni
ng, I
nc.
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?
3. Complete the table.
Number of sides of regular polygon 3 4 5 6 7 8
Sum of measures of interior angles
Measure of each interior angle
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
Chapter 8 ! Assignments 143
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
Assignment Assignment for Lesson 8.4
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°
Chapter 8 ! Assignments 143
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
Assignment Assignment for Lesson 8.4
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°
44
144 Chapter 8 ! Assignments
© 2
010
Car
negi
e Le
arni
ng, I
nc.
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
Lesson 8.6 | Quadrilateral Family 467
© 2
010
Car
negi
e Le
arni
ng, I
nc.
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.
46
True or False
Lesson 8.6 | Quadrilateral Family 469
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
Lesson 8.6 | Quadrilateral Family 469
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
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
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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?
50
More Interior Exterior Angles
Chapter 8 ! Assessments 1205
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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.
Determine whether each statement is true or false. If it is false, explain why.
9. If the diagonals of a quadrilateral are congruent, then the quadrilateral is a square.
51
1210 Chapter 8 ! Assessments
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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
52
Chapter 8 ! Assessments 1211
© 2
010
Car
negi
e Le
arni
ng, I
nc.
8
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