GEORGIA MATHEMATICS 1 - Carnegie Learning

© 2008 Carnegie Learning, Inc. 

GEORGIA 

MATHEMATICS 1 

Teacher’s Resources 

and Assessments


Pre-Test 

Name ___________________________________________________ Date _____________________ 

1. In the figure shown below, PW � NC, PA � NK, and AC � KW. 

Determine whether 

�PAW is congruent to �NKC. Use complete sentences to justify your reasoning. 

2. In the figure shown below, �RAC � �YTC. Determine which angle is congruent to �CRA. 

R 

P 

E 

A C W K 

A C 

N 

T 

Y 

Georgia Mathematics 1 Chapter 7 ■ Assessments 161 

7

7 

Pre-Test PAGE 2 

3. In the figure shown below, DG � GE 

and �GDP � �GEH. 

Determine whether �GDP is 

congruent to �GEH. Use complete sentences to justify your reasoning. 

E 

4. Using the figure shown below, determine whether �TAR is congruent to �PAR. 

Use complete sentences to justify your reasoning. 

T 

D 

A 

F 

P 

H 

R P 

162 Chapter 7 ■ Assessments Georgia Mathematics 1 

G 

© 2008 Carnegie Learning, Inc.



Name ___________________________________________________ Date _____________________ 

5. Sketch two triangles with appropriate markings that are congruent by using 

HL Congruence Postulate. 

6. In �CAT and �DQG, AC � QD, 

CT � DG, 

and �A � �Q. 

Name one additional piece of 

information that is needed to prove �CAT � �DQG? 

Use complete sentences to justify 

your reasoning. 


7

7 




Post-Test 

Name ___________________________________________________ Date _____________________ 

1. In the figure shown below, PA � NK and NA � PK. 

Determine whether �PAN is 

congruent to �NKP. Use complete sentences to justify your reasoning. 

A 

2. Using the figure shown below, determine whether �RAC is congruent to �YTC. 

Use complete 

sentences to justify your reasoning. 

R 

P N 

E 

A 

K 

C 

T 


Y 

7

7 

Post-Test PAGE 2 

3. In the figure shown below, DF � EF 

and �GHE � �GPD. 


�DFH � �EFP. Use complete sentences to justify your reasoning. 

E 

D 

4. Using the figure shown below, �T � �P and AR bisects TP. Determine whether 

�TAR � �PAR. Use complete sentences to justify your reasoning. 

A 

P 

H 

T R 

P 

F 


G 




Name ___________________________________________________ Date _____________________ 

5. In the isosceles trapezoid shown below, AV � RM and Q is the midpoint of VM. 

Determine whether �RAQ � �ARQ. Use complete sentences to justify your reasoning. 

A R 

V Q M 

6. Sketch �BAG and �REG such that BA � RE, �B � �R, and they are congruent by 

using AAS Congruent Postulate. 


7

7 




Mid-Chapter Test 

Name ___________________________________________________ Date _____________________ 

1. In parallelogram HPJM, HP || MJ 

and MP is a diagonal. Determine whether 

�HPM � �JMP. 


2. In the figure shown below, AN bisects the vertex angle of isosceles triangle YAW. 

Determine whether AN is a segment bisector of YW. Use complete sentences to justify 


A 

Y N W 


7

7 

Mid-Chapter Test PAGE 2 

3. Using the figure shown below, determine whether �R � �Y. 

Use complete sentences to 

justify your reasoning. 

R 

4. Triangle ABC is congruent to triangle XYZ. Name all pairs of corresponding parts. 

5. In the figure shown below, quadrilateral ABCD is a rectangle with diagonal DB. 

Determine whether �ABD � �CDB. Use complete sentences to justify your reasoning. 

A B 

D 

A 

C 

C 

T 


Y 



End of Chapter Test 

Name ___________________________________________________ Date _____________________ 

1. Name all of the pairs of congruent triangles in the figure drawn below and state why each 

pair of triangles is congruent. 

R 

2. In the figure shown below, PE � RA and PA � RE. 

Determine whether quadrilateral PARE 

is a parallelogram. Use complete sentences to justify your reasoning. 

P A 

E 

I 

T 

R 

G 

H 


7

7 

End of Chapter Test PAGE 2 

3. Using the figure shown below, determine whether �HAW � �TAW. 

Use complete 


W 

H 

T 

4. In the figure shown below, CQ � WQ, 

and HC � HW. 


�CHQ � �WHQ. 


H 

Q 

A 

C W 





Name ___________________________________________________ Date _____________________ 

5. In the figure shown below, BD bisects AC and AB � BC. 


�ADB and �CDB are right triangles. Use complete sentences to justify your reasoning. 

A 

6. In the figure shown below, QP � AP and AY � QY. 

Determine whether �Q � �A. 


R 

W 

K 

B 

D 

P 

C 

Y 

Q 

A 


7

7 


7. In the figure shown below, �J � �X and JD � XD. 


Determine whether BD � CD. 

B C 

D 

J X 




Standardized Test Practice 

Name ___________________________________________________ Date _____________________ 

1. Which of these cannot be used to prove that two triangles are congruent? 

a. AA 

b. SAS 

c. AAS 

d. HL 

Use the figure below to answer Questions 2 and 3. In the figure, GM � MO 

and EO � MO. 

G E 

M 

2. �GMO � �EOM by: 

a. SAS 

b. AA 

c. HL 

d. Not enough information is given to determine if the triangles are congruent 

3. �MTG � �MTO by: 

a. SSS 

b. AA 

c. SAS 

T 



O 

7

7 

Standardized Test Practice PAGE 2 

Use the figure shown below to answer Questions 4 through 6. 

4. Given: �BUN is isosceles with BU � BN. 

What additional given information is needed to 

prove �BUG � �BNA by ASA? 

a. 

b. 

c. 

d. 

5. Triangle BAG is isosceles with BG � BA. 

What is one additional piece of information that 

is needed to prove �BGN � �BAU by SAS? 

a. 

b. 

c. 

d. 

�UBA � �NBG 

�UBG � �NBA 

�BGU � �BAN 

�BUG � �BNA 

GN � AU 

BU � BN 

GU � AN 

AG � GA 

B 

U G A N 

6. If �UBG � �NBA, 

why is �UBA � �NBG? 

a. Reflexive Property 

b. Transitive Property 

c. Angle addition 

d. Definition of congruent angles 





Name ___________________________________________________ Date _____________________ 

7. Points G, E, and T form a triangle. GE � TR, 

GP � TQ, 

and PE � QR. 

Which triangle is 

�GEP congruent to? 

a. 

b. 

c. 

d. 

�RTQ 

�QTR 

�TQR 

�TRQ 

8. Using the diagram shown at the right, choose the 

correct conclusion. 

a. The triangles are congruent by SSS. 

b. The triangles are congruent by ASA. 

c. The triangles are congruent by SAS. 

d. Not enough information to determine if the triangles 

are congruent. 








10. In the diagram shown at the right, PN is parallel to JQ, 

KN � 16 centimeters, PN � 8 centimeters, and JQ � 14 centimeters. 

Find the length of NQ. 

a. 28 cm 

b. 12 cm 

c. 9.1 cm 

d. 7 cm 


R 

A 

20 0 

P 

J 

C 

K 

20 0 

T 

N 

Y 

Q 

7


7 

11. In the figure shown below, �ABD and �BCD are isosceles triangles and �ABD � �CBD. 

�ABD � �CBD by: 

A 

a. SSS 

b. SAS 

c. AAS 

d. HL 

12. In the figure shown below, PS � RQ. �SQR � �QSP by: 

P Q 

a. SSS 

b. SAS 

c. AAS 

d. HL 

B 

D 

S 

C 

R 




Pre-Test 

Name ___________________________________________________ Date _____________________ 

1. In the figure shown below, PW � NC, PA � NK, and AC � KW. 


�PAW is congruent to �NKC. Use complete sentences to justify your reasoning. 

A C W K 

2. In the figure shown below, �RAC � �YTC. Determine which angle is congruent to �CRA. 

R 

P 

E 

A C 

�CRA � �CYT 

N 

Sample Answer: It is given that PW � NC 

and PA � NK. AW � KC because AC � KW 

and CW is added to each to form the triangle side. Therefore, �PAW � �NKC by 

SSS Congruence Postulate. 

T 

Y 


7

7 


3. In the figure shown below, DG � GE and �GDP � �GEH. 

Determine whether �GDP is 

congruent to �GEH. Use complete sentences to justify your reasoning. 

E 

4. Using the figure shown below, determine whether �TAR is congruent to �PAR. 


T 

D 

A 

F 

P 

H 

Sample Answer: Angle G is shared by both triangles. One other pair of angles and pair of 

included sides are also congruent, so �GDP � �GEH 

by ASA Congruence Postulate. 

R P 

Sample Answer: In the figure, TR � PR. Side AR is shared by both triangles. All right 

angles are congruent, so �TRA � �PRA. Therefore, �TAR � �PAR by SAS Congruence 

Postulate. 


G 




Name ___________________________________________________ Date _____________________ 

5. Sketch two triangles with appropriate markings that are congruent by using 


Sample Answer: 

6. In �CAT and �DQG, AC � QD, 

CT � DG, 

and �A � �Q. 

Name one additional piece of 

information that is needed to prove �CAT � �DQG? 

Use complete sentences to justify 


Sample Answer: The triangles are congruent using ASA Congruence Postulate if 

�C � �D. The triangles are congruent using SAS Congruence Postulate if AT � QG. 


7

7 




Post-Test 

Name ___________________________________________________ Date _____________________ 

1. In the figure shown below, PA � NK and NA � PK. Determine whether �PAN is 

congruent to �NKP. Use complete sentences to justify your reasoning. 

A 

2. Using the figure shown below, determine whether �RAC is congruent to �YTC. 

Use complete 


R 

P N 

E 

A 

K 

Sample Answer: Side PN is shared by both triangles. The other two pairs of 

corresponding sides are equal, so �PAN � �NKP by SSS Congruence Postulate. 

C 

T 

Sample Answer: In the figure, AR � TY 

and AC � TC. Angle RCA and angle YCT are 

vertical angles, so �RCA � �YCT. So, two pairs of sides and a pair of non-included 

angles are known. Because the SAS Congruence Postulate must use the included angles, 

there is not enough information to determine whether the triangles are congruent. 


Y 

7

7 


3. In the figure shown below, DF � EF and �GHE � �GPD. 


�DFH � �EFP. Use complete sentences to justify your reasoning. 

E 

D 

4. Using the figure shown below, �T � �P and AR bisects TP. Determine whether 

�TAR � �PAR. Use complete sentences to justify your reasoning. 

A 

P 

H 

T R 

P 

F 

Sample Answer: Supplements of congruent angles are congruent, so �DHF � �EPF. 

Vertical angles are congruent, so �DFH � �EFP. �DFH � �EFP by AAS Congruence 

Postulate. 

Sample Answer: If �T � �P then �TAP is isosceles and AT � AP. By definition of bisect, 

TR � PR. So, �TAR � �PAR by SAS Congruence Postulate. 


G 




Name ___________________________________________________ Date _____________________ 

5. In the isosceles trapezoid shown below, AV � RM and Q is the midpoint of VM. 

Determine whether �RAQ � �ARQ. Use complete sentences to justify your reasoning. 

A R 

V Q M 

Sample Answer: Base angles of an isosceles trapezoid are equal, so �V � �M. 

So, 

�VAQ � �MRQ by SAS Congruence Postulate. Corresponding parts of congruent 

triangles are congruent, so AQ � RQ. Triangle AQR is isosceles and base angles of an 

isosceles triangle are congruent, so �RAQ � �ARQ. 

6. Sketch �BAG and �REG such that BA � RE, �B � �R, and they are congruent by 

using AAS Congruent Postulate. 

Sample Answer: 

B 

A 

G 

E 

R 


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7 




Mid-Chapter Test 

Name ___________________________________________________ Date _____________________ 

1. In parallelogram HPJM, HP || MJ and MP is a diagonal. Determine whether 

�HPM � �JMP. 


Sample Answer: Alternate interior angles formed by parallel lines are congruent, 

so �HPM � �JMP 

and �PMH � �MPJ. Side MP is shared by both triangles. 

So, �HPM � �JMP by ASA Congruence Postulate. 

2. In the figure shown below, AN bisects the vertex angle of isosceles triangle YAW. 

Determine whether AN is a segment bisector of YW. Use complete sentences to justify 


A 

Y N W 

Sample Answer: By definition of bisect, �YAN � �WAN. By definition of isosceles 

triangle, AY � AW. Side AN is shared by both triangles. Therefore, �YAN � �WAN by 

SAS Congruence Postulate. Corresponding parts of congruent triangles are congruent, 

so YN � WN. By definition of segment bisector, AN is a segment bisector of YW. 


7

7 

Mid-Chapter Test PAGE 2 

3. Using the figure shown below, determine whether �R � �Y. 

Use complete sentences to 

justify your reasoning. 

R 

4. Triangle ABC is congruent to triangle XYZ. Name all pairs of corresponding parts. 

5. In the figure shown below, quadrilateral ABCD is a rectangle with diagonal DB. 

Determine whether �ABD � �CDB. Use complete sentences to justify your reasoning. 

A B 

D 

A 

C 

C 

T 

Sample Answer: In the figure, AR � TY and AC � TC. Angle RCA and �YCT are vertical 

angles, so �RCA � �YCT. So, two pairs of sides and a pair of non-included angles are 

known. Because the SAS Congruence Postulate must use the included angles, there is not 

enough information to determine whether the triangles are congruent and it is also 

unknown whether the corresponding angles, �R and �Y, 


�A � �X 

�B � �Y 

�C � �Z 

AB � XY 

BC � YZ 

AC � XZ 

Sample Answer: Side BD is shared by both triangles. By definition of rectangle, AB � CD 

and AD � CB. So, �ABD � �CDB by SSS Congruence Postulate. 


Y 



End of Chapter Test 

Name ___________________________________________________ Date _____________________ 

1. Name all of the pairs of congruent triangles in the figure drawn below and state why each 

pair of triangles is congruent. 

R 

2. In the figure shown below, PE � RA and PA � RE. 

Determine whether quadrilateral PARE 

is a parallelogram. Use complete sentences to justify your reasoning. 

P A 

E 

I 

T 

R 

G 

H 

�GHI � �TIH by HL Congruence Postulate, �TIH � �ITR by SAS Congruence Postulate, 

and �GHI � �ITR by the Transitive Property. 

Side AE 

is shared by both triangles. So, �PAE � �REA by SSS Congruence Postulate. 

Corresponding parts of congruent triangles are congruent, so �EAR � �AEP and 

�EAP � �AER. 

If alternate interior angles are congruent, then lines are parallel, so 

AR || PE and PA || ER. Opposite sides are parallel, so quadrilateral ABCD is a 

parallelogram. 


7

7 


3. Using the figure shown below, determine whether �HAW � �TAW. Use complete 


W 

H 

T 

4. In the figure shown below, CQ � WQ, 

and HC � HW. 


�CHQ � �WHQ. 


H 

Q 

A 

Sample Answer: Side WA 

is shared by both triangles. So, �HAW � �TAW by 


C W 

Sample Answer: Side HQ is shared by both triangles. So, �CHQ � �WHQ by SSS 

Congruence Postulate. Corresponding parts of congruent triangles are congruent, 

so �CHQ � �WHQ. 





Name ___________________________________________________ Date _____________________ 

5. In the figure shown below, BD bisects AC and AB � BC. 


�ADB and �CDB are right triangles. Use complete sentences to justify your reasoning. 

A 

6. In the figure shown below, QP � AP and AY � QY. 



R 

W 

K 

B 

D 

P 

C 

Sample Answer: By definition of bisect, AD � CD. Side BD is shared by both triangles. 

So, �ABD � �CBD by SSS Congruence Postulate. Corresponding parts of congruent 

triangles are congruent, so �ADB � �CDB. Angle ADB and �CDB are congruent and 

supplementary so each measures 90º. By definition, �ADB and �CDB are right angles. 

Y 

Q 

A 

�Q � �A. 

Sample Answer: Draw PY, 

which is shared by both triangles. So, �PYQ � �PYA by 

SSS Congruence Postulate. Corresponding parts of congruent triangles are congruent, 

so �Q � �A. 


7

7 


7. In the figure shown below, �J � �X and JD � XD. 


Determine whether BD � CD. 

B C 

D 

J X 

Sample Answer: Vertical angles are congruent, so �JDB � �XDC. 

So, �JDB � �XDC by 

ASA Congruence Postulate. Corresponding parts of congruent triangles are congruent, so 

BD � CD. 




Standardized Test Practice 

Name ___________________________________________________ Date _____________________ 

1. Which of these cannot be used to prove that two triangles are congruent? 

a. AA 

b. SAS 

c. AAS 

d. HL 

Use the figure below to answer Questions 2 and 3. In the figure, GM � MO 

and EO � MO. 

G E 

M 

2. �GMO � �EOM by: 

a. SAS 

b. AA 

c. HL 


3. �MTG � �MTO by: 

a. SSS 

b. AA 

c. SAS 

T 



O 

7

7 


Use the figure shown below to answer Questions 4 through 6. 

4. Given: �BUN is isosceles with BU � BN. 

What additional given information is needed to 

prove �BUG � �BNA by ASA? 

a. 

b. 

c. 

d. 

5. Triangle BAG is isosceles with BG � BA. 

What is one additional piece of information that 

is needed to prove �BGN � �BAU by SAS? 

a. 

b. 

c. 

d. 

�UBA � �NBG 

�UBG � �NBA 

�BGU � �BAN 

�BUG � �BNA 

GN � AU 

BU � BN 

GU � AN 

AG � GA 

B 

U G A N 

6. If �UBG � �NBA, 

why is �UBA � �NBG? 

a. Reflexive Property 

b. Transitive Property 

c. Angle addition 

d. Definition of congruent angles 





Name ___________________________________________________ Date _____________________ 

7. Points G, E, and T form a triangle. GE � TR, 

GP � TQ, 

and PE � QR. 

Which triangle is 

�GEP congruent to? 

a. 

b. 

c. 

d. 

�RTQ 

�QTR 

�TQR 

�TRQ 















10. In the diagram shown at the right, PN is parallel to JQ, 

KN � 16 centimeters, PN � 8 centimeters, and JQ � 14 centimeters. 

Find the length of NQ. 

a. 28 cm 

b. 12 cm 

c. 9.1 cm 

d. 7 cm 


R 

A 

20 0 

P 

J 

C 

K 

20 0 

T 

N 

Y 

Q 

7


7 

11. In the figure shown below, �ABD and �BCD are isosceles triangles and �ABD � �CBD. 

�ABD � �CBD by: 

A 

a. SSS 

b. SAS 

c. AAS 

d. HL 

12. In the figure shown below, PS � RQ. �SQR � �QSP by: 

P Q 

a. SSS 

b. SAS 

c. AAS 

d. HL 

B 

D 

S 

C 

R 




Assignment 

Assignment for Lesson 7.1 

Name ___________________________________________________ Date _____________________ 

Glass Lanterns 

Introduction to Congruence 

1. Summarize what you know about similar figures and dilations. 

2. Summarize what you know about the relationship between congruent figures and isometries. 

Use the hexagons below to answer Questions 3 and 4. 

A 

Sample Answer: All dilations involve figures that are similar to one another. Most similar 

figures are dilations of one another. The exception is similar figures that are also 

congruent. 

Sample Answer: All congruent figures are isometries of one another and all isometries 

describe congruent figures. 

B 

E 

D 

C 

J 

I 

F 

G 

H 

3. Use a ruler and protractor to determine whether the two hexagons are similar. If the 

figures are similar, write a similarity statement about them. Use a complete sentence to 

explain your answer. 

Sample Answer: Hexagon ABCDE is similar to hexagon HIJFG. The corresponding angles 

are congruent, and corresponding sides are proportional. So, ABCDE ~ HIJFG. 

4. Are the two hexagons congruent? If they are congruent, write a congruency statement 

describing the relationship. Use a complete sentence to explain your answer. 

Sample Answer: Because ABCDE ~ HIJFG 

and the ratio of corresponding sides is 1, 

ABCDE � HIJFG. 

Georgia Mathematics 1 Chapter 7 ■ Assignments 101 

7

7 

Use the figures below to answer Questions 5 and 6. 

A 

5. Use a ruler and protractor to determine whether the triangles are similar. If they are 

similar, write a similarity statement. Use a complete sentence to explain your answer. 

6. Are the triangles congruent? If they are congruent, write a congruency statement. 

Use a complete sentence to explain your answer. 

7. In the figure below, �EMA � �OMP. 

Name the pairs of corresponding angles and name 

the pairs of corresponding sides. 

E 

A 

B 

M 

C 

O 

�E and �O, �A and �P, �EMA and �OMP 

Z 

Sample Answer: �ABC ~ �YXZ because corresponding angles are congruent and 

corresponding pairs of sides have the same ratio. 

P 

X 

Sample Answer: �ABC is not congruent to �YXZ because the ratio of corresponding 

sides is not 1. 

EA and OP, EM and OM, AM and PM 

102 Chapter 7 ■ Assignments Georgia Mathematics 1 

Y 



Name ___________________________________________________ Date _____________________ 

8. In the figure below, �EMA is similar, but not congruent, to �OMP. Name the pairs of 

corresponding angles and name the pairs of corresponding sides. 

E 

A 

EA and OP, EM 

M P 

�E and �O, �A and �P, �EMA and �OMP 

O 

and OM, AM and PM 

9. Draw two congruent scalene triangles that share only a common vertex. Name the pairs 

of corresponding angles and name the pairs of corresponding sides. Write a congruence 

statement for your triangles. 

Answers will vary. Sample Answer: 

A 

�A and �E, �B and �D, �ACB and �ECD, 

AB and ED, 

BC and DC, AC and EC 

�ABC � �EDC 

E 

B C D 


7

7 

10. Draw two congruent right triangles that share only a common side. Name the pairs of 

corresponding angles and name the pairs of corresponding sides. Write a congruence 

statement for your triangles. 

Answers will vary. Sample Answer: 

X 

Y 

Z 

W 

�X and �Z, �XWY and �ZWY, �WYX and �WYZ 

XW and ZW, XY and ZY, WY and WY 

�XYW � �ZYW 




Assignment 

Name ___________________________________________________ Date _____________________ 

Computer Graphics 

Proving Triangles Congruent: SSS and SAS 

Complete the statements below about triangle congruence. 

1. If you know that the ____________________ corresponding sides of two triangles are congruent, then the 

triangles are congruent by the __________________________. 

SSS Theorem 

2. If you know that two pairs of __________________________ 

corresponding sides 

are congruent, and the 

_____________________ included angles are congruent, then you know that triangles are congruent by the 

____________________________. 

SAS Theorem 

In Questions 3 and 4, prove that the triangles are congruent. 

3. 

VT � CB; VD � CG; TD � GB 

V 

T 

D 

B C 

VT � CB, VD � CG, and 

DT � GB. 

G 

Assignment for Lesson 7.2 

So, �VTD � �CBG by the SSS Congruence Theorem. 


7

7 

4. 

Read the scenario below. Use the scenario to complete Question 5. 

The figure below is a basic plan for a decorative porch roof. For construction purposes, 

�DPA � �DPG. 

You know from your construction that DP � AG and DP bisects AG. 

B 

AB � RE; m�B � m�E; BC � EW 

A 

C 

D 

C 

A P G 

E 

F 

B 

5. Can you prove �DPA � �DPG? 

Complete the two-column proof. 

Statement Reason 

W 

R 

AB � RE, BC � EW, 

and m�B � m�E. So, �ABC � �REW by the SAS Congruence 

Theorem. 

1. DP � AG 

1. Given ___________________________________________ 

2. �DPA and �DPG are right angles. 2. Perpendicular ___________________________________________ 

lines intersect to form right angles 

3. �DPA � �DPG 

3. All ___________________________________________ 

right angles are congruent 

4. AP � GP 

4. Definition ___________________________________________ 

of bisect 

5. DP � DP 

5. Reflexive ___________________________________________ 

Property 

6. �DPA � �DPG 

6. SAS ___________________________________________ 

Congruence Theorem 


E

GEORGIA MATHEMATICS 1 - Carnegie Learning

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