draw the quadrilateral family venn diagram with all the...
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Honors Geometry Chapter 5 Practice Problem Answers
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1.
Draw the Quadrilateral Family Venn Diagram with all the associated definitions and properties.
Honors Geometry Chapter 5 Practice Problem Answers
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2.
3.
5.
4.
3.
2.
1. Both pairs of opposite sides of a quadrilateral are parallel
Both pairs of opposite sides of a quadrilateral are congruent
One pair of opposite sides of a quadrilateral are both parallel andcongruent
The diagonals of a quadrilateral bisect each other
Both pairs of opposite angles of a quadrilateral are congruent
Write 5 ways to prove that a quadrilateral is a parallelogram:
Diagonals are not ≅ Diagonals are ≅
No right ∠s At least 1 right ∠ (actually, all ∠s are right!)
All 4 sides are ≅ Not all 4 sides are ≅ (only opposite sides are ≅)
Diagonals bisect the ∠s Diagonals do not bisect the ∠s
Diagonals are not ⊥Diagonals are ⊥
Rectangle that is not a RhombusRhombus that is not a Rectangle
1.
2.
3.
4.
5.
Name 5 properties of a rectangle that is not a rhombus and the corresponding propertiesof a rhombus that is not a rectangle:
Honors Geometry Chapter 5 Practice Problem Answers
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4.
5.
Diagonals bisect each other and bisect the ∠s Rhombus
Diagonals are ⊥ and ≅ and bisect each other Square
Consecutive ∠s measure 30°, 150°, 110°, 70° Trapezoid
Consecutive sides measure 15, 18, 18, 15 Kite
ParallelogramConsecutive sides measure 15, 18, 15, 181.
2.
3.
4.
5.
Give the best name for a quadrilateral whose:
If 2 ∠s of a trapezoid are ≅, the trapezoid is isosceles. Sometimes
If a parallelogram is equilateral, it is equiangular. Sometimes
If the diagonals of a quadrilateral divide each ∠ such that each halfmeasures 45°, the quadrilateral is a square.
Always
SometimesIf the diagonals of a quadrilateral are ≅, it is an isosceles trapezoid.1.
2.
3.
4.
Answer Sometimes, Always, or Never
Honors Geometry Chapter 5 Practice Problem Answers
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6.
7.
x+24( ) + 2x+6( ) = 180
⇒ x = 50
⇒m∠C = m∠A = 2 50( ) + 6 = 106°
x+24
2x+6
ABCD is a parallelogram. Find the measure of ∠C.
DA
B C
x x+6( ) = 160
⇒ x2 + 6x - 160 = 0
⇒ x + 16( ) x - 10( ) = 0
⇒x = -16 and/or x = 10
Can't be -16 as this makes AD < 0
∴ x = 10 and PABCD = 52 u
x
x+6
ABCD is a rectangle. The area is 160 u2. Find the perimeter.
C
A
D
B
Honors Geometry Chapter 5 Practice Problem Answers
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8.
9.
9. OSP is isosceles 9. Definition of Isosceles
OSP is isosceles
ONR PS
Prove:
Given:
Converse of ITT8.8. OP ≅ OS7. ∠P ≅ ∠S 7. Transitive Property of ≅ ∠s6. ∠ORN ≅ ∠S 6. PCA
Statements Reasons
5.4.3.2.1. Given
All radii of a are ≅ITTGivenPCA5. ∠ONR ≅ ∠P
3. ∠ONR ≅ ∠ORN4. NR PS
2. ON ≅ OR1. O
S
N
O
P
R
∠A supp. ∠B
∠C supp. ∠D
Prove:
Given:
Statements Reasons
3.2.
1. Given
SSISPPSSIS3. ∠A supp. ∠B
2. AD BC1. ∠C supp. ∠D
A
B
D
C
Honors Geometry Chapter 5 Practice Problem Answers
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10.
11.
YZ bisects ∠AYB
CY ≅ AYYZ CA
Prove:
Given:
7. YZ bisects ∠AYB 7. Definition of ∠ bisector
6. ∠ZYB ≅ ∠AYZ 6. Transitive Property of ≅ ∠s
Statements Reasons
5.4.3.
2.1. Given
ITT
GivenPCAPAI5. ∠CAY ≅ ∠AYZ
3. YZ CA4. ∠C ≅ ∠ZYB
2. ∠C ≅ ∠CAY1. CY ≅ AY
Z
C
A
Y
B
6. DC AB 6. Given
S
S
A
11. AD ≅ BC 11. CPCTCSAS 2, 8, 4( )10.10. DOA ≅ COBTransitive Property of ≅ ∠s9.9. ∠DOA ≅ ∠COB
8. ∠OCD ≅ ∠COB 8. PAI7. ∠ODC ≅ ∠DOA 7. PAI
Given:
Prove:
ODC AB
AD ≅ BC
Statements Reasons
5.4.3.2.1. Given
All radii of a are ≅Auxiliary LinesAll radii of a are ≅ITT5. ∠ODC ≅ ∠OCD
3. Draw OD & OC4. OD ≅ OC
2. OA ≅ OB1. O
C
BOA
D
Honors Geometry Chapter 5 Practice Problem Answers
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12.
13.
S
S
S
9. DPC & APB are isosceles 9. Definition of Isosceles
Subtraction Property of ≅ Segments8.8. AP ≅ BP
7. PD ≅ PC 7. Converse of ITT6. ∠PDC ≅ ∠PCD 6. CPCTC
Given:
Prove:
ABCD is an isoscelestrapezoid with legs AD & BC
DPC & APB are isosceles
Statements Reasons
5.4.
3.2.
1. Given
Definition of Legs of Isos. TrapezoidDiagonals of an Isos. Trap. are ≅
Reflexive PropertySSS 2, 3, 4( )5. ADC ≅ BCD
3. AC ≅ BD
4. DC ≅ DC
2. AD ≅ BC
1. ABCD is an isosceles trapezoid with legs AD & BC
P
C
A B
D
10. AC bisects EF 10. Definition of Segment Bisector
A
S
A
CPCTC9.9. GF ≅ GEASA 3, 7, 4( )8.8. AFG ≅ CEG
7. AF ≅ EC 7. Subtraction Property of ≅ Segments6. BE ≅ DF 6. Given
Given:
Prove:
ABCD is a BE ≅ DF
AC bisects EF
Statements Reasons
5.4.3.2.1. Given
Definition of a PAIPAIOpposite sides of a are ≅5. AD ≅ BC
3. ∠DAC ≅ ∠BCA4. ∠AFE ≅ ∠CEF
2. AD BC1. ABCD is a
G
F
CB
A D
E
Honors Geometry Chapter 5 Practice Problem Answers
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14.
15.
A
AS
CPCTC9.9. GH ≅ EFASA 5, 4, 7( )8.8. AGH ≅ CFE
7. ∠DCA ≅ ∠BAC 7. PAI6. DC AB 6. Definition of
Given:
Prove:
ABCD is a ∠GHA ≅ ∠FECHB ≅ DE
GH ≅ EF
Statements Reasons
5.4.3.2.1. Given
Opposite sides of a are ≅GivenSubtraction Property of ≅ SegmentsGiven5. ∠GHA ≅ ∠FEC
3. HB ≅ DE4. AH ≅ EC
2. DC ≅ AB1. ABCD is a
F
E
BA
D C
H
G
Given:
Prove:
ID bisects BRBI ≅ IR
BIRD is a kite
Statements Reasons
5.4.
3.
2.
1. Given
Definition of Segment Bisector
Given
E.T. 2,3( )If one diagonal of a quadrilateral isthe ⊥ bis. of the other, then thequadrilateral is a kite
5. BIRD is a kite
3. BI ≅ IR
4. ID ⊥ bis. BR
2. BK ≅ KR
1. ID bisects DR
K
D
I
B R
Honors Geometry Chapter 5 Practice Problem Answers
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16.
17.
A
AS
10. YTWX is a rhombus 10. If a has 2 consecutive sides ≅, itis a rhombus
CPCTC9.9. YT ≅ WTASA 5, 6, 7( )8.8. YTP ≅ WTZ
7. ∠YTP ≅ ∠WTZ 7. Reflexive Property6. TP ≅ TZ 6. Given
Given:
Prove:
YTWX is a YP ⊥ TWZW ⊥ TYTP ≅ TZ
YTWX is a rhombus
Statements Reasons
5.4.3.2.1. Given
GivenGivenDefinition of ⊥ segmentsRAT5. ∠YPT ≅ ∠WZT
3. ZW ⊥ TY4. ∠YPT & ∠WZT are right ∠s
2. YP ⊥ TW1. YTWX is a
Z
P W
Y
T
X
SAA
10. FBCE is a 10. If a quadrilateral has both pairs ofopposite sides ≅, it is a
Subtraction Property of ≅ Segments9.9. FE ≅ BCCPCTC8.8. AB ≅ ED
7. AC ≅ FD 7. Opposite sides of a are ≅6. FB ≅ CE 6. CPCTC
Given:
Prove:
ACDF is a ∠AFB ≅ ∠ECD
FBCE is a
Statements Reasons
5.4.3.2.1. Given
Opposite sides of a are ≅GivenOpposite ∠s of a are ≅ASA 3, 2, 4( )5. AFB ≅ DCE
3. ∠AFB ≅ ∠ECD4. ∠A ≅ ∠D
2. AF ≅ DC1. ACDF is a
E
CA
F D
B
Honors Geometry Chapter 5 Practice Problem Answers
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18.
19.
5. AC & BD bisect each other
9. DEC is isosceles 9. Definition of Isosceles Division Property of ≅ Segments8.8. ED ≅ ECThe diagonals of a rectangle are ≅7.7. AC ≅ BD
6. E is the midpoint of AC & BD 6. Definition of Segment Bisector
4. ABCD is a rectangle 4. A with at least one right ∠ is arectangle
Given:
Prove:
ABCD is a AB ⊥ BC
DEC is isosceles
Statements Reasons
3.
1.
5.
2. Given
The diags of a bisect each other
Given
Definition of ⊥ segments3. ∠ABC is right
1. ABCD is a 2. AB ⊥ BC
E
C
A
B
D
8. ABCD is a rectangle
6. ABCD is a parallelogram
4. AEB ≅ DEC3. ∠AEB ≅ ∠DEC
1. AED & BEC are isosceles with≅ bases AD & CB, respectively
A2. AE ≅ ED; BE≅EC
7. AC ≅ DB
5. AB ≅ DC
S S
If a parallelogram has ≅ diagonals, itis a rectangle
8.7. Addition Property of ≅ Segments
6. If a quadrilateral has two pairs ofopposite sides ≅, it is a parallelogram
Given:
Prove:
AED & BEC are isosceles with≅ bases AD & BC, respectively
ABCD is a rectangle
Statements Reasons
5.4.3.2.
1. Given
Definition of Isosceles VATSAS 2, 3, 2( )CPCTC
E
C
A
B
D
Honors Geometry Chapter 5 Practice Problem Answers
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20.
21.
Prove that the quadrilateral formed by connecting the midpoints of the sides of a is a .
A ASS
S S3. E, F, G, & H are midpoints
A quadrilateral with both pairs ofopposite sides ≅ is a
8.8. ABCD is a 7. EH ≅ GF; HG ≅ FE 7. CPCTC6. AEH ≅ CGF; DHG ≅ BFE 6. SAS 4, 5, 4( )
Given:
Prove:
ABCD is a E, F, G, & H are midpoints
EFGH is a
Statements Reasons
5.
4.3.2.1. Given
The opposite sides of a are ≅GivenDivision Property of ≅ Segments
The opposite ∠s of a are ≅5. ∠A ≅ ∠C; ∠B ≅ ∠D
4. AE ≅ CG; AH ≅ CF; HD ≅ FB; EB ≅ GD
2. AB ≅ DC; AD ≅ BC1. ABCD is a
F
G
H
E
C
A
B
D
3. ∠W ≅ ∠X
6. RWXY is an isosceles trapezoid 6. A trapezoid with lower base ∠scongruent is isosceles
Given:
Prove:
TWX is isosceles with base WXRY WX
RWXY is an isoscelestrapezoid
Statements Reasons
5.4.
3.2.
1. Given
Definition of Isosceles ITT
GivenDefinition of Trapezoid5. RWXY is a trapezoid
4. RY WX
2. TW ≅ TX
1. TWX is isosceles with base WX
Y
T
W X
R
Honors Geometry Chapter 5 Practice Problem Answers
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22.
23.
A A
SSS S
3. AE ≅ BF ≅ CG ≅ DH
10. ABCD is a 10. A quadrilateral with both pairs ofopposite sides ≅ is a
CPCTC9.9. AB ≅ CD; AD ≅ CBSAS 3, 7, 4( )8.8. AFB ≅ CHD; BGC ≅ DEA
7. ∠DEA ≅ ∠BGC; ∠AFB ≅ ∠CHD 7. Supps. of ≅ ∠s are ≅
6. ∠EFG supp. ∠AFB; ∠GHE supp. ∠CHD; ∠HEF supp. ∠DEA; ∠FGH supp. ∠BGC
6. If 2 ∠s form a straight ∠, they are supp.
Given:
Prove:
EFGH is a AE ≅ BF ≅ CG ≅ DH
ABCD is a
Statements Reasons
5.4.3.2.1. Given
The opposite sides of a are ≅GivenAddition Property of ≅ SegmentsThe opposite ∠s of a are ≅5. ∠FEH ≅ ∠HGF; ∠GHE ≅ ∠EFG
4. AF ≅ CH; DE ≅ BG
2. EF ≅ GH; HE ≅ FG1. EFGH is a
D
CB
GE
F
HA
10. RM PO 10. AIP
S
S
A
3. MT ≅ PS
11. ABCD is a 11. A quadrilateral with one pair of oppositesides both & ≅ is a
CPCTC9.9. ∠TMR ≅ ∠SPOCPCTC8.8. RM ≅ PO
7. RTM ≅ OSP 7. SAS 3, 5, 6( )6. RT ≅ SO 6. The opposite sides of a are ≅
Given:
Prove:
RSOT is a parallelogramMS ≅ TP
MOPR is a parallelogram
Statements Reasons
5.4.3.2.1. Given
GivenAddition Property of ≅ SegmentsDefinition of PAI5. ∠RTS ≅ ∠OST
4. RT SO
2. MS ≅ TP
1. RSOT is a
P
O
R
S
T
M
Honors Geometry Chapter 5 Practice Problem Answers
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24.
15. SA bisects ∠PSR 15. Definition of ∠ bisector
14. ∠RSA ≅ ∠PSA 14. Transitive Property of ≅ ∠s13. ∠RAS ≅ ∠PSA 13. PAI
6. ∠QAP ≅ ∠QPA 6. Transitive Property of ≅ ∠s
12. ∠RAS ≅ ∠RSA 12. ITT11. RA ≅ RS 11. Transitive Property of ≅ Segments
Definition of Midpoint10.10. QA ≅ RAGiven9.9. A is the midpoint of QR
8. QP ≅ RS 8. Opposite sides of a are ≅7. QA ≅ QP 7. Converse of ITT
Given:
Prove:
PQRS is a A is the midpoint of QRPA bisects ∠QPS
SA bisects ∠PSR
Statements Reasons
5.4.3.2.
1. Given
GivenDefinition of ∠ bisectorDefinition of PAI5. ∠QAP ≅ ∠SPA
3. ∠QPA ≅ ∠SPA4. QR PS
2. PA bisects ∠QPS
1. PQRS is a
S
RAQ
P
Honors Geometry Chapter 5 Practice Problem Answers
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25.
13. JMP is equilateral 13. Definition of Equilateral
6. JP KO 6. Definition of
12. JM ≅ MP ≅ JP 12. Addition Property of ≅ Segments11. JK ≅ KM ≅ MO ≅ OP ≅ JR ≅ RP 11. Transitive Property of ≅ Segments
Opposite sides of a are ≅10.10. JK ≅ RO; JR ≅ KO
Definition of 7.7. JKOR is a
9. RP ≅ KO; OP ≅ KR 9. Opposite sides of a are ≅8. KM ≅ RO; MO ≅ KR 8. Opposite sides of a are ≅
Given:
Prove:
KOR is equilateralKOPR is a KMOR is a
JMP is equilateral
Statements Reasons
5.4.3.2.1. Given
Definition of Equilateral GivenDefinition of Given5. KOPR is a
3. KMOR is a 4. JM RO
2. KO ≅ KR ≅ RO1. KOR is equilateral
P
J
M O
K R