Download Unit 6 - Congruent Triangles Congruent Triangles Classwork 1

Unit 6 - Congruent Triangles
Congruent Triangles
Classwork
1. Given that ABC  XYZ, identify and mark all of the congruent corresponding parts
in the diagram.


2. CAT  JSD. List each of the following.
a. three pairs of congruent sides

b. three pairs of congruent angles
For exercises 3 – 5 list the corresponding sides and angles. Write a congruence statement.
3.
4.
5.
For Exercises 6 and 7, can you conclude that the triangles are congruent?
Justify your answers.
6. GHJ and IHJ


Geometry-Congruent Triangles
7. QRS and TVS


~1~
NJCTL.org
8. If ACB  JKL, which of the following must be a correct congruence statement?
A.

A  L
C. AB  JL

B. B  K
D. BAC  LKJ
9. A student says she can use the information in the figure to prove  ACB   CAD.
Is she correct? Explain.


10. Use the information given in the diagram and the Reasons Bank to give a reason why
each statement is true. Some reasons may be used more than once.
Statements
Reasons




a.  L   Q
a.
b.  LNM   QNP
b.
c.  M   P
c.

d. LM  QP, LN  QN , MN  PN
e.LNM  QNP


d.
e.
Reasons Bank:
All corresponding parts are congruent, so
triangles are congruent.
Vertical angles are congruent
Given
Third Angles Theorem
Geometry-Congruent Triangles
~2~
NJCTL.org
Congruent Triangles
Homework
11. Given that DEF  JKL, mark all of the congruent corresponding parts in the
diagram and then list them.
D

J

E
F
L
K
12. BAT  COM. List each of the following.
a. three pairs of congruent sides


b. three pairs of congruent angles
For exercises 13 – 15 list the corresponding sides and angles. Write a congruence statement.
13.
14.
15.
For Exercises 16 and 17, can you conclude that the triangles are congruent?
Justify your answers.
16. SRT and PRQ
17. ABC and FGH
18. If PLM  DOB, which of the following must be a correct congruence statement?
A.

C.
D  L
B. B  M
PM  OB
D. LM  DO

Geometry-Congruent Triangles
~3~
NJCTL.org
19. A student says she can use the information in the figure to prove  JLK  JLM.
Is she correct? Explain.


A
20. Use the information given in the diagram and the Reasons Bank to give
a reason why each statement is true. Some reasons may be used more
than once.
B
C
Given: AD and BE bisect each other. AB  DE ; ∠A ≅ ∠D
Prove: ∆ACB ≅ ∆DCE
E
Statements
1) AD and BE bisect each other.
AB  DE , A  D
Reasons
1) Given
2) AC  DC , BC  EC
4) B  E
2)
____________________________
3)
_____________________
4)
5) ACB  DCE
5)
3) ACB  DCE
D
Reasons Bank:
Third Angles Theorem
Definition of a bisector
Vertical angles are congruent
All corresponding parts are congruent, so
triangles are congruent.
Geometry-Congruent Triangles
~4~
NJCTL.org
Proving Congruence (Triangle Congruence: SSS and SAS)
Classwork
Given MGT to answer questions 21 – 23.
21. What angle is included between GM and MT ?

22. Which sides include ∠T?
23. What angle is included between GT and MG ?
24. What additional information is needed to prove the two triangles congruent by SAS
Triangle Congruence?
Are the triangles congruent? If so, state the congruence postulate and write a congruence
statement. If there is not enough information to prove the triangles congruent, write not
enough information.
25.
28.
31.
Geometry-Congruent Triangles
26.
27.
29.
30.
32.
33.
~5~
NJCTL.org
Proving Congruence (Triangle Congruence: SSS and SAS)
Homework
Given PFK to answer questions 34 – 36.
34. What angle is included between PF and PK?

35. Which sides include ∠F?
36. What angle is included between FK and KP?
37. What additional information is needed to prove the two triangles congruent by SSS
Triangle Congruence?
Are the triangles congruent? If so, state the congruence postulate and write a
congruence statement. If there is not enough information to prove the triangles
congruent, write not enough information.
38.
41.
44.
39.
40.
42.
45.
Geometry-Congruent Triangles
11.
43.
46.
~6~
12.
NJCTL.org
13.
Proving Congruence (Triangle Congruence: ASA, AAS and HL)
Classwork
If ABC ≅ ∆ XYZ by the given theorem, what is the missing congruent part? Draw and
mark a diagram.
47. ASA Triangle 

48. ASA Triangle 
A  X
AB  XY
49. ASA Triangle 
ZY  CB
Y  B
AC  XZ
C  Z
For numbers 50 – 56, if the triangles are congruent, state which theorem applies and
write the congruence statement.
50.
53.
51.
52.
54.
55.
56.
Geometry-Congruent Triangles
~7~
NJCTL.org
Proving Congruence (Triangle Congruence: ASA, AAS, and HL)
Homework
If ∆PLK ≅ ∆YUO by the given postulate or theorem, what is the missing congruent part?
Draw and mark a diagram.
57. ASA Triangle 
58. AAS Triangle 
K  O
PK  YO
59. ASA Triangle 
LP  UY
Y  P
U  L
K  O
For numbers 60 – 66, if the triangles are congruent, state which theorem applies and
write the congruence statement.
60.
61.
62.
∆EFG, ∆GHF
63.
64.
65.
66.
Geometry-Congruent Triangles
~8~
NJCTL.org
Congruent Triangle Proofs – CP
Classwork
PARCC-type problems
Complete the two-column proof with the reasons bank provided.
Some reasons may be used more than once & some may not be
used at all.
67. Given: BC  DC, AC  EC
Prove: ABC ≅ EDC
Reasons Bank:
Third Angles Theorem
Definition of a bisector
Vertical angles are congruent
Given
SSS Triangle Congruence
SAS Triangle Congruence
ASA Triangle Congruence
AAS Triangle Congruence
HL Triangle Congruence
Statements
Reasons
1. 𝐵𝐶 ≅ 𝐷𝐶 , 𝐴𝐶 ≅ 𝐸𝐶
2. ∠𝐵𝐶𝐴 ≅ ∠𝐷𝐶𝐸
3. ABC ≅ EDC
68. Given: ∠K ≅ ∠M, KL ≅ ML
Prove: ∆JKL ≅ ∆PML
Reasons Bank:
Third Angles Theorem
Definition of a bisector
Vertical angles are congruent
Given
SSS Triangle Congruence
SAS Triangle Congruence
ASA Triangle Congruence
AAS Triangle Congruence
HL Triangle Congruence
1. ∠𝐾 ≅ ∠𝑀, 𝐾𝐿 ≅ 𝑀𝐿
2. ∠𝐽𝐿𝐾 ≅ ∠𝑃𝐿𝑀
3. ∆JKL ≅ ∆PML
69. Given: LOM  NPM,
LM  NM
Prove: ∆LOM  ∆NPM
Reasons Bank:
Third Angles Theorem
Definition of a bisector
Vertical angles are congruent
Given
SSS Triangle Congruence
SAS Triangle Congruence
ASA Triangle Congruence
AAS Triangle Congruence
HL Triangle Congruence
Geometry-Congruent Triangles
1. ∠𝐿𝑂𝑀 ≅ ∠𝑁𝑃𝑀, 𝐿𝑀 ≅ 𝑁𝑀
2. ∠𝐿𝑀𝑂 ≅ ∠𝑁𝑀𝑃
3. ∆LOM  ∆NPM
~9~
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Congruent Triangle Proofs – CP
Homework
PARCC-type problems
Complete the two-column proof with the reasons bank provided.
Some reasons may be used more than once & some may not be
used at all.
70. Given: WX || YZ ,WX  YZ
Prove: WXZ  YZX
Reasons Bank:
If two parallel lines are cut by a transversal, 1. 𝑊𝑋 || 𝑌𝑍, 𝑊𝑋 ≅ 𝑌𝑍
then the corresponding angles are
2. ∠𝑊𝑋𝑍 ≅ ∠𝑌𝑍𝑋
congruent
3. 𝑋𝑍 ≅ 𝑋𝑍
If two parallel lines are cut by a transversal,
4. ∆𝑊𝑋𝑍 ≅ ∆𝑌𝑍𝑋
then the alternate interior angles are
congruent
Reflexive Property of Congruence
Transitive Property of Congruence
Given
ASA Triangle Congruence
SSS Triangle Congruence
AAS Triangle Congruence
SAS Triangle Congruence
HL Triangle Congruence
71.
Reasons Bank:
If two parallel lines are cut by a transversal,
then the corresponding angles are
congruent
1. ∠𝑄 ≅ ∠𝑆, ∠𝑇𝑅𝑆 ≅ ∠𝑇𝑅𝑄
If two parallel lines are cut by a transversal,
2. 𝑅𝑇 ≅ 𝑇𝑅
then the alternate interior angles are
3. ∆𝑄𝑇𝑅 ≅ ∆𝑆𝑅𝑇
congruent
Reflexive Property of Congruence
Transitive Property of Congruence
Given
SSS Triangle Congruence
SAS Triangle Congruence
ASA Triangle Congruence
AAS Triangle Congruence
HL Triangle Congruence
72. Given: HIJ  KIJ
IJH  IJK
Prove: ∆HIJ  ∆KIJ
Reasons Bank:
If two parallel lines are cut by a transversal,
then the corresponding angles are
congruent
If two parallel lines are cut by a transversal, 1. ∠𝐻𝐼𝐽 ≅ ∠𝐾𝐼𝐽, ∠𝐼𝐽𝐻 ≅ ∠𝐼𝐽𝐾
̅ ≅ 𝐽𝐼
̅
2. 𝐽𝐼
then the alternate interior angles are
congruent
3. ∆𝐻𝐼𝐽 ≅ ∆𝐾𝐼𝐽
Reflexive Property of Congruence
Transitive Property of Congruence
Given
ASA Triangle Congruence
SSS Triangle Congruence
AAS Triangle Congruence
SAS Triangle Congruence
HL Triangle Congruence
Geometry-Congruent Triangles
~10~
NJCTL.org
Congruent Triangle Proofs – Honors
Classwork
PARCC-type problems
Write a two-column proof.
67. Given: BC  DC, AC  EC
Prove: ABC ≅ EDC
Statements
Reasons
68. Given: ∠K ≅ ∠L, KL ≅ LM
Prove: ∆JKL ≅ ∆PML
69. Given: LOM  NPM,
LM  NM
Prove: ∆LOM  ∆NPM
Geometry-Congruent Triangles
~11~
NJCTL.org
Congruent Triangle Proofs – Honors
Homework
PARCC-type problems
Write a two-column proof.
70. Given: WX || YZ ,WX  YZ
Prove: WXZ  YZX
71.
72. Given: HIJ  KIJ
IJH  IJK
Prove: ∆HIJ  ∆KIJ
Geometry-Congruent Triangles
~12~
NJCTL.org
CPCTC – CP
Classwork
For numbers 73 – 74 state the reason the two triangles are congruent. Then list all other corresponding
parts of the triangles that are congruent.
74. Given: HI  JG
73.
PARCC-type problems
75. Complete the proof with the statements/reasons bank provided.
Some statements/reasons may be used more than once & some
may not be used at all.
Given: GK is the perpendicular bisector of FH .
Prove: FG  HG
Statements
Reasons
1) GK is the perpendicular bisector of FH . 1)
2)
2) Def. of perpendicular bisector
3) GKF  GKH
3) All right
4)
4) Reflexive Prop. of 
5) ∆FGK  ∆HGK
5)
6)
6) CPCTC
are .
nt
Geometry-Congruent Triangles
~13~
NJCTL.org
76. Given: YA  BA , B  Y
Prove: AZ  AC
Statements
Reasons
1. ?
1. ?
2. ?
2. Vertical angles are congruent
3. ∆𝑌𝑍𝐴 ≅ ∆𝐵𝐶𝐴
3. ?
4. ?
4. ?
Statements/Reasons Bank:
𝐴𝑍 ≅ 𝐴𝐶
𝑌𝑍 ≅ 𝐵𝐶
Given
SSS Triangle Congruence
SAS Triangle Congruence
ASA Triangle Congruence
AAS Triangle Congruence
HL Triangle Congruence
Geometry-Congruent Triangles
CPCTC
Vertical angles are congruent
Reflexive property of ≅
𝑌𝐴 ≅ 𝐵𝐴
∠𝑍 ≅ ∠𝐶
∠𝐵 ≅ ∠𝑌
∠𝑍𝐴𝑌 ≅ ∠𝐶𝐴𝐵
Transitive property of ≅
~14~
NJCTL.org
CPCTC – CP
Homework
For numbers 77 – 78 state the reason the two triangles are congruent. Then list all other corresponding
parts of the triangles that are congruent.
77. ∆ZXW and ∆YWX
78. ∆ABE and ∆ACD
PARCC-type problems
79. Complete the proofs with the statements/reasons bank provided.
Some statements/reasons may be used more than once & some
may not be used at all.
Given: ABCE is a rectangle; D is the midpoint of CE .
Prove: AD  BD
Statements
1) ABCE is a rectangle. D is
the midpoint of CE .
Reasons
1)
2) AED  BCD
2) Definition of rectangle
3) AE  BC
3) Definition of rectangle
4) 𝐷𝐸 ≅ 𝐷𝐶
4)
5) ∆𝐴𝐸𝐷 ≅ ∆𝐵𝐶𝐷
5)
6)
6)
Statements/Reasons Bank:
𝐴𝐵 ≅ 𝐴𝐵
𝐴𝐷 ≅ 𝐵𝐷
Given
SSS Triangle Congruence
SAS Triangle Congruence
ASA Triangle Congruence
AAS Triangle Congruence
HL Triangle Congruence
Geometry-Congruent Triangles
CPCTC
Vertical angles are congruent
Reflexive Property of ≅
𝐷𝐸 ≅ 𝐷𝐶
∠𝐸𝐴𝐷 ≅ ∠𝐶𝐵𝐷
∠𝐴𝐷𝐸 ≅ ∠𝐵𝐷𝐶
∠𝐴𝐷𝐵 ≅ ∠𝐴𝐷𝐵
Definition of Midpoint
~15~
NJCTL.org
80. Given: BD  AC , D is the midpoint of AC .
Prove: BC  BA
Statements
Reasons
1) ?
1) ?
2) 𝐴𝐷 ≅ 𝐶𝐷
2) ?
3) 𝐵𝐷 ≅ 𝐵𝐷
3) ?
4) ?
4) Perpendicular lines form right angles
5) ∠𝐴𝐷𝐵 ≅ ∠𝐶𝐷𝐵
5) ?
6) ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷
6) ?
7) ?
7) ?
Statements/Reasons Bank:
CPCTC
𝐶𝐴 ≅ 𝐴𝐶
Given
SSS Triangle Congruence
SAS Triangle Congruence
ASA Triangle Congruence
AAS Triangle Congruence
HL Triangle Congruence
Geometry-Congruent Triangles
Vertical angles are congruent
All right angles are congruent
Reflexive property of ≅
𝐵𝐶 ≅ 𝐵𝐴
∠𝐴 ≅ ∠𝐶
∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷
∠𝐴𝐷𝐵 & ∠𝐶𝐷𝐵 are right angles
Transitive property of ≅
~16~
NJCTL.org
CPCTC – Honors
Classwork
For numbers 73 – 74 state the reason the two triangles are congruent. Then list all other corresponding
parts of the triangles that are congruent.
74. Given: HI  JG
73.
PARCC-type problems
75. Complete the proof.
Given: GK is the perpendicular bisector of FH .
Prove: FG  HG
Statements
Reasons
1) GK is the perpendicular bisector of FH . 1)
2)
2) Def. of perpendicular bisector
3) GKF  GKH
3) All right
4)
4) Reflexive Prop. of 
5) ∆FGK  ∆HGK
5)
6)
6) CPCTC
Geometry-Congruent Triangles
~17~
are .
NJCTL.org
76. Write a proof.
Given: YA  BA , B  Y
Prove: AZ  AC
Statements
Geometry-Congruent Triangles
Reasons
~18~
NJCTL.org
CPCTC – Honors
Homework
For numbers 77 – 78 state the reason the two triangles are congruent. Then list all other corresponding
parts of the triangles that are congruent.
77. ∆ZXW and ∆YWX
78. ∆ABE and ∆ACD
PARCC-type problems
79. Complete the proof.
Given: ABCE is a rectangle; D is the midpoint of CE .
Prove: AD  BD
Statements
Reasons
1) Given
1) ABCE is a rectangle. D is
the midpoint of CE .
2) AED  BCD
2) Definition of rectangle
3) AE  BC
3) Definition of rectangle
4)
4)
5)
5)
6)
6)
Geometry-Congruent Triangles
~19~
NJCTL.org
80. Write a proof.
Given: BD  AC , D is the midpoint of AC .
Prove: BC  BA
Statements
Geometry-Congruent Triangles
Reasons
~20~
NJCTL.org
Isosceles and Equilateral Triangles
Classwork
Complete the statement using ALWAYS, SOMETIMES, and NEVER.
81. An isosceles triangle is ___________ a scalene triangle.
82. An equilateral triangle is __________ an isosceles triangle.
83. An isosceles triangle is ___________ an equilateral triangle.
84. An acute triangle is ___________ an equiangular triangle.
85. An isosceles triangle is __________ a right triangle.
Solve for each variable in exercises 86 – 94. Figures are not drawn to scale.
86.
87.
88.
89.
90.
91.
92.
93.
Geometry-Congruent Triangles
94.
~21~
NJCTL.org
Isosceles and Equilateral Triangles
Homework
Complete the statement using ALWAYS, SOMETIMES, and NEVER.
95. A scalene triangle is ___________ an equilateral triangle.
96. An equilateral triangle is __________ an obtuse triangle.
97. An isosceles triangle is ___________ an acute triangle.
98. An equiangular triangle is ___________ a right triangle.
99. A right triangle is __________ an isosceles triangle.
Solve for each variable in exercises 100 – 108. Figures are not drawn to scale
100.
101.
103.
104.
106.
Geometry-Congruent Triangles
102.
105.
107.
108.
~22~
NJCTL.org
Congruent Triangles - Unit Review
PMI Geometry
Multiple Choice – Circle the correct answer
1.
In the given triangle, find x and y.
a. x = 32, y = 5
b. x = 5, y = 116°
c. x = 5, y = 32°
d. x = 5, y = 64°
x
32°
y°
5
32°
2. If ∆𝐷𝐸𝐹 ≅ ∆𝑃𝑄𝑅, one set of corresponding sides are:
a. 𝐷𝐸 , 𝑄𝑅
b. 𝐸𝐹 , 𝑃𝑄
c. 𝐷𝐸 , 𝑃𝑄
d. 𝐷𝐹 , 𝑅𝑄
3. If ∆𝐺𝐻𝐼 ≅ ∆𝐽𝐾𝐿, which of the following must be a correct congruence statement?
a. ∠𝐺 ≅ ∠𝐿
b. 𝐺𝐻 ≅ 𝐾𝐿
c. 𝐺𝐼 ≅ 𝐽𝐾
d. ∠𝐻 ≅ ∠𝐾
4. Given ∆𝑀𝑁𝑂, which angle is included between 𝑀𝑁 & 𝑀𝑂?
a. ∠𝑁𝑀𝑂
b. ∠𝑀𝑁𝑂
c. ∠𝑁𝑂𝑀
d. ∠𝑀𝑂𝑁
5. Given ∆𝑋𝑌𝑍, which side is included between ∠𝑍𝑋𝑌 & ∠𝑌𝑍𝑋?
a. 𝑋𝑌
b. 𝑌𝑍
c. 𝑋𝑍
d. 𝑌𝑋
6. Are the triangles congruent – if so, by which congruence postulate/theorem?
a. SAS
Q
S
b. ASA
c. AAS
d. Not congruent
V
U
P
R
7. By which postulate/theorem, if any, are the two triangles congruent?
a. ASA
c. SAS
b. AAS
d. Not congruent
Geometry-Congruent Triangles
~23~
NJCTL.org
8. State the third congruence needed to make ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 true by SAS congruence.
Given:
∠B ≅ ∠E
𝐴𝐵 ≅ 𝐷𝐸
a.
b.
c.
d.
𝐴𝐶 ≅
𝐵𝐶 ≅
∠C ≅
∠A ≅
𝐷𝐹
𝐸𝐹
∠F
∠D
9. What information must be true for ASA congruence between the two triangles?
H
K
a. 𝐻𝐼 ≅ 𝐾𝐿
28°
28°
b. 𝐺𝐻 ≅ 𝐽𝐾
G 95°
c. ∠I ≅ ∠L
95°
J
̅
d. 𝐺𝐼 ≅ 𝐽𝐿
I
L
10. State the third congruence needed to make ∆𝑋𝑌𝑍 ≅ ∆𝑃𝑄𝑅 true by ASA congruence.
Given:
∠P ≅ ∠X
∠Y ≅ ∠Q
a.
b.
c.
d.
XY ≅ PQ
PQ ≅ 𝑌𝑍
∠X ≅ ∠P
XZ ≅ PR
Short Constructed Response – Write the correct answer for each question.
No partial credit will be given.
#11- 12 For the triangles in the diagram:
 list the corresponding parts
 list the congruence postulate or theorem, if any
 write a congruence statement, if any
11.
A
A
12.
B
X
M
X
N
C
D
I
13. Find the value of each variable in the figure below.
(x + 5)°
7z°
(2x)° (12y + 2)°
Geometry-Congruent Triangles
~24~
NJCTL.org
Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given.
14. Fill in the proof below using the “Reason
H
Reasons Bank
Bank” off to the right. Some reasons may
be used more than once and some may
not be used at all.
Given: 𝐻𝐼 ⊥ 𝐺𝐽, 𝐺𝐻 ≅ 𝐽𝐻
Prove: I is the midpoint of 𝐺𝐽
Statements
1.) 𝐻𝐼 ⊥ 𝐺𝐽
2.) ∠𝐻𝐼𝐺 & ∠𝐻𝐼𝐽 are right angles
3.) ∠𝐻𝐼𝐺 ≅ ∠𝐻𝐼𝐽
4.) 𝐺𝐻 ≅ 𝐽𝐻
5.) 𝐻𝐼 ≅ 𝐻𝐼
6.) ∆HIG ≅ ∆HIJ
7.) 𝐺𝐼 ≅ 𝐽𝐼
8.) I is the midpoint of 𝐺𝐽
G
Reasons
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
I
J
SSS
SAS
ASA
AAS
HL
CPCTC
Def. of perpendicular lines
Def. of midpoint
All right angles are congruent
Given
Vertical angles are congruent
Reflexive Property of ≅
Transitive Property of ≅
Symmetric Property of ≅
The base angles of an isosceles
triangle are ≅
Honors:
15. Write a two-column or flow proof.
Given: 𝑀𝑁 ≅ 𝑀𝑋, ∠𝐼 ≅ ∠𝐴
Prove: 𝑁𝐼 ≅ 𝑋𝐴
N
A
M
I
X
Geometry-Congruent Triangles
~25~
NJCTL.org
Unit 5 - Congruent Triangles - ANSWER KEY
Congruent Triangles
Classwork
b.) CB
c.) YXZ
1.
a. ) XY
2.
a.) CA  JS , AT  SD, CT  JD
d.) AC
e.) C
f.) Y
b.) C  J, A  S, T  D
3. Sides: ML  PO , LN  OQ, MN  PQ
Angles: L  O, M  P, N  Q
Congruence Statement: MLN  POQ
4. Sides: ST UW , RS  VU, RT  VW
 T W,
 S  U
Angles: R  V,
Congruence Statement: RTS  VWU
5. Sides: CA  XZ ,BC  YX, AB  ZY
(Since the triangles are isosceles, other answers may be correct.)
 C 
Angles: B  Y,
X, A  Z
Congruence Statement: BCA  YXZ
6. Yes; all corresponding sides and angles are congruent.
7. No; there are no congruent
sides


8. C
9. Yes; B  D by the Third Angle Theorem and AC  AC by the reflexive property of congruence.
10. a.) Given
b.) Vertical Angles are congruent
c.) Third Angle Theorem
d.) Given
e.) All corresponding parts are congruent, so triangles are congruent.
Geometry-Congruent Triangles
~26~
NJCTL.org
Congruent Triangles
Homework
11.
DE  JK , EF  KL, DF  JL, D  J, E  K, F  L
12.
a.) BA  CO , AT  OM, BT  CM
b.) B  C, A  O, T  M
13. Sides: LS  RZ, LP  RH, SP  ZH
Angles: L  R, S  Z, P  H
Congruence Statement: SLP  ZRH
14. Sides: AC FQ , BC  DQ, AB  FD
Angles: B  
D, A 
F, C  Q
Congruence Statement: ACB  FQD
15. Sides: WQ  YT , QE  TR, WE  YR
Y, Q 
Angles: W  
T, E  R
Congruence Statement: WQE  YTR
16. No; there are no congruent sides
17. Yes; A  F bythe Third
Angle Theorem so all corresponding sides and angles are congruent.

18. B
19. Yes; K  M by the Third Angle Theorem and JL  JL by the reflexive property of congruence.
20. 1.) Given
2.) Definition of a bisector
d.) Third Angle Theorem
Geometry-Congruent Triangles
c.) Vertical Angles are congruent
e.) All corresponding parts are congruent, so triangles are congruent.
~27~
NJCTL.org
Proving Congruence (Triangle Congruence: SSS and SAS)
Classwork
21. M
22. GT and TM
23. G
24. B  G
25. SSS Triangle Congruence; ABC  DFE
26. SAS Triangle Congruence; GIH  JHI

MKL  NPO
27. SAS Triangle Congruence;


28. Not enough information


29. SAS Triangle Congruence;
WZV  XZY
30. SSS Triangle Congruence; ABD  CDB

31. Not enough information


JMK  LMK
32. SAS Triangle Congruence;
33. SAS Triangle Congruence; ONQ  RQN


Proving Congruence (Triangle Congruence: SSS and SAS)


Homework
34. P
35. PF and FK
36. K
37. XY  RS
38. SSS Triangle Congruence; YQE  WQE
39. Not enough information (SSA does not work)

RSE  PTJ
40. SAS Triangle Congruence;
41. Not enough information

Geometry-Congruent Triangles

~28~
NJCTL.org
42. Not enough information
43. SSS Triangle Congruence; BAD  BCD (Since the triangles are isosceles, other statements
may be true.)
44. SAS Triangle Congruence;

GFE  HFI
45. Not enough information
Congruence;

46. SSS/SAS Triangle
JIL  LKJ

Geometry-Congruent Triangles

~29~
NJCTL.org
Proving Congruence (Triangle Congruence: ASA, AAS and HL)
Classwork
Y
B
Y
B
#48
47. B  Y
48. A  X
C
A
Z
X
A
#49
C
X
B
Z
Y
49. A  X
C
A
50. ASA Triangle Congruence; TUV  WXY
Z
X
51. Not enough information
52. AAS Triangle Congruence;

WXY  AZY
53. HL Triangle Congruence; PMN  NOP
54. ASA/AAS Triangle
Congruence;
 TUV  WXV
55. AAS Triangle Congruence;

ZAB  CED
56. HL Triangle Congruence;
 KLN
 KMN


Proving Congruence
 (Triangle
 Congruence: ASA, AAS & HL)
U
Homework
L
L
U
#58
57. P  Y
K
P
O
Y
K
P
58. K  O
U
L
59. KL  OU
#59
60. AAS Triangle Congruence; BCE  DCF
P
O
Y
K
Y
O
61. ASA Triangle Congruence; PSQ  RQS
62. HL Triangle Congruence;

EFG  HFG

63. AAS Triangle Congruence;
TUV  YWX


64. Not enough information


65. ASA Triangle Congruence;

STV  UVT
66. HL Triangle Congruence; TUX  VWX


Geometry-Congruent Triangles


~30~
NJCTL.org
Congruent Triangle Proofs – both CP & Honors have the same answers
Classwork
67.
Statements
68.
1. BC  DC; AC  EC
1. Given
2. BCA  DCE
2. Vertical Angles are congruent
3. ABC  EDC
3. SAS Triangle Congruence
Statements


69.
1. Given
2. JLK  PLM
2. Vertical Angles are congruent
3. JKL  PML
3. ASA Triangle Congruence


Reasons____________
1. K  M ; KL  ML
Statements

Reasons____________
Reasons____________
1. LOM  NPM ; LM  NM
1. Given
2. LMO  NMP
2. Vertical Angles are congruent
3. LOM  NPM
3. AAS Triangle Congruence

Congruent Triangle Proofs – both CP & Honors have the same answers
Homework
70.
Statements
Reasons____________
1. WX || YZ ; WX  YZ
1. Given
2. WXZ  YZX
2. If 2 parallel lines are cut by a transversal, then the
alternate interior angles are congruent
3. XZ  XZ
3. Reflexive Property of Congruence
4. WXZ  YZX
4. SAS Triangle Congruence
Statements
Reasons____________

71.



1. Q  S ; TRS  TRQ
1. Given
2. RT  TR
2. Reflexive Property of Congruence
3. QTR  SRT
3. AAS Triangle Congruence

Geometry-Congruent Triangles
~31~
NJCTL.org
72.
Statements

Reasons____________
1. HIJ  KIJ ; IJH  IJK
1. Given
2. JI  JI
2. Reflexive Property of Congruence
3. HIJ  KIJ
3. ASA Triangle Congruence

CPCTC – both CP & Honors have the same answers
Classwork
73. AAS Theorem; H  M, JK  KL, HK  KM
74. HL Theorem; HG  JI, GJH  IHJ, IJH  GHJ
75.
Statements
Reasons_________________
1. Given
2. FK  HK & ∠𝐺𝐾𝐹 & ∠𝐺𝐾𝐻 are right ∡𝑠
4. GK  GK
5. SAS Triangle Congruence
6. FG  HG
76.
Statements

Reasons____________
1. B  Y ; YA  BA
1. Given
2. ZAY  CAB
2. Vertical Angles are congruent
3. YZA  BCA
3. ASA Triangle Congruence
4. AZ  AC
4. CPCTC


Geometry-Congruent Triangles
~32~
NJCTL.org
CPCTC – both CP & Honors have the same answers
Homework
77. SAS Postulate; ZX  YW, Z  Y, ZXW  YWX
78. ASA Postulate; DC  EB, AC  AB, C  B
79.

Statements
Reasons_________________
4. DE  DC
4. Definition of Midpoint
5. AED  BCD
5. SAS Triangle Congruence
6. AD  BD
6. CPCTC

80.
Statements

Reasons____________
1. BD ⊥ AC ; D is midpt. of AC
1. Given
2. AD  CD
2. Definition of Midpoint
3. BD  BD
3. Reflexive Property of Congruence
4. ADB and CDB are rt. ∠s
4. Perpendicular lines form right angles
5. ADB  CDB
5. All right angles are congruent
6. ABD  CBD
6. SAS Triangle Congruence
7. BC  BA
7. CPCTC


Geometry-Congruent Triangles
~33~
NJCTL.org
Isosceles and Equilateral Triangles
Classwork
81. never
82. always
83. sometimes
84. sometimes
85. sometimes
86. x = 72
87. x = 5; y = 74
88. x = 3; y = 60
89. x = 42; y = 96; z = 21
90. x = 66; y = 57
91. m = 83; u = 106; x = 14; y = 60; z = 60
92. x = 11
93. x = 48; y = 84; z = 4
94. x = 10; y = 20
Isosceles and Equilateral Triangles
Homework
95. never
96. never
97. sometimes
98. never
99. sometimes
100. x = 108
101. x = 3; y = 63
102. x = 2; y = 60; z = 60
103. x = 74; y = 148; z = 16
104. x = 9; y = 64; z = 64
105. m = 37; u = 106; x = 37; y = 106; z = 106
106. x = 7
107. x = 45; y = 45
108. x = 10; y = 60
Geometry-Congruent Triangles
~34~
NJCTL.org
Unit Review Answer Key
1. b
2. c
3. d
4. a
5. c
6. b
7. c
8. b
9. b
10. a
11. SAS
∠A ≅ ∠D, ∠B ≅ ∠C, ∠AXB ≅ ∠DXC
𝐴𝐵 ≅ 𝐷𝐶 , 𝐴𝑋 ≅ 𝐷𝑋 , 𝑋𝐵 ≅ 𝑋𝐶
∆AXB ≅ ∆DXC
12. AAS
∠A ≅ ∠I, ∠X ≅ ∠N, ∠AMX ≅ ∠IMN
𝐴𝑀 ≅ 𝐼𝑀 , 𝐴𝑋 ≅ 𝐼𝑁 , 𝑀𝑋 ≅ 𝑀𝑁
∆AMX ≅ ∆IMN
13. x = 35, y = 9 & z = 5
14. Statements
1.) 𝐻𝐼 ⊥ 𝐺𝐽
2.) ∠𝐻𝐼𝐺 & ∠𝐻𝐼𝐽 are right angles
3.) ∠𝐻𝐼𝐺 ≅ ∠𝐻𝐼𝐽
4.) 𝐺𝐻 ≅ 𝐽𝐻
5.) 𝐻𝐼 ≅ 𝐻𝐼
6.) ∆HIG ≅ ∆HIJ
7.) 𝐺𝐼 ≅ 𝐽𝐼
8.) I is the midpoint of 𝐺𝐽
15.
Statement
𝑀𝑋 ≅ 𝑀𝑁
∠I ≅ ∠A
∠NMI ≅ ∠XMA
∆NMI ≅ ∆XMA
𝑁𝐼 ≅ 𝑋𝐴
Geometry-Congruent Triangles
Reasons
1.) Given
2.) Def. of perpendicular lines
3.) All right angles are congruent
4.) Given
5.) Reflexive Property of Congruence
6.) HL
7.) CPCTC
8.) Definition of Midpoint
Reason
given
given
vertical angles are congruent
AAS
CPCTC
~35~
NJCTL.org
Geometry-Congruent Triangles
~36~
NJCTL.org