Draw the Quadrilateral Family Venn Diagram with all the …doversherborn.comcastbiz.net/highschool/academics/math... · 2015-11-01 · Draw the Quadrilateral Family Venn Diagram with

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.



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:



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



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



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



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



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



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



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



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



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



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



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



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

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Draw the Quadrilateral Family Venn Diagram with all the …doversherborn.comcastbiz.net/highschool/academics/math... · 2015-11-01 · Draw the Quadrilateral Family Venn Diagram with