Geometry-Congruent Triangles ~1~ NJCTL.org 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 7. QRS and TVS
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Geometry-Congruent Triangles ~1~ NJCTL.org
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 7.
QRS and
TVS
Geometry-Congruent Triangles ~2~ NJCTL.org
8. If
ACB
JKL, which of the following must be a correct congruence statement?
A. A L B. B K
C. AB JL 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 d.
e.
LNM
QNP e.
Reasons Bank: All corresponding parts are congruent, so
triangles are congruent. Vertical angles are congruent Given Third Angles Theorem
Geometry-Congruent Triangles ~3~ 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. D L B. B M
C. PM OB D. LM DO
Geometry-Congruent Triangles ~4~ NJCTL.org
19. A student says she can use the information in the figure to prove
JLK
JLM. Is she correct? Explain.
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.
Given: AD and BE bisect each other. AB DE ; ∠A ≅ ∠D
Prove: ∆ACB ≅ ∆DCE
Statements Reasons
1) AD and BE bisect each other.
AB DE , A D
1) Given
2) AC DC , BC EC 2) _________________________________________________ 3) ACB DCE 3)
4) B E 4)
5) ACB DCE
5)
Reasons Bank: Third Angles Theorem Definition of a bisector Vertical angles are congruent All corresponding parts are congruent, so
triangles are congruent.
D
C
B
E
A
Geometry-Congruent Triangles ~5~ 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. 26. 27.
28. 29. 30. 31. 32. 33.
Geometry-Congruent Triangles ~6~ 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.
41. 42. 43.
11. 12. 13.
38. 39. 40.
44. 45. 46.
Geometry-Congruent Triangles ~7~ NJCTL.org
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.
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 ~9~ 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.
68. Given: ∠K ≅ ∠M, KL ≅ ML
Prove: ∆JKL ≅ ∆PML
69. Given: LOM NPM,
LM NM
Prove: ∆LOM ∆NPM
67. Given: ,BC DC AC EC
Prove: ABC ≅ EDC
Statements Reasons
1. 𝐵𝐶 ≅ 𝐷𝐶 , 𝐴𝐶 ≅ 𝐸𝐶
2. ∠𝐵𝐶𝐴 ≅ ∠𝐷𝐶𝐸
3. 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
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
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. ∠𝐿𝑂𝑀 ≅ ∠𝑁𝑃𝑀, 𝐿𝑀 ≅ 𝑁𝑀
2. ∠𝐿𝑀𝑂 ≅ ∠𝑁𝑀𝑃
3. ∆LOM ∆NPM
Geometry-Congruent Triangles ~10~ NJCTL.org
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.
71.
72. Given: HIJ KIJ
IJH IJK
Prove: ∆HIJ ∆KIJ
70. Given: || , WX YZ WX YZ
Prove: WXZ YZX
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, then the alternate interior angles are congruent
Reflexive Property of Congruence Transitive Property of Congruence
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, then the alternate interior angles are congruent
Reflexive Property of Congruence Transitive Property of Congruence
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, then the alternate interior angles are congruent
Reflexive Property of Congruence Transitive Property of Congruence
1. 𝑊𝑋 || 𝑌𝑍 ,𝑊𝑋 ≅ 𝑌𝑍
2. ∠𝑊𝑋𝑍 ≅ ∠𝑌𝑍𝑋
3. 𝑋𝑍 ≅ 𝑋𝑍
4. ∆𝑊𝑋𝑍 ≅ ∆𝑌𝑍𝑋
1. ∠𝑄 ≅ ∠𝑆, ∠𝑇𝑅𝑆 ≅ ∠𝑇𝑅𝑄
2. 𝑅𝑇 ≅ 𝑇𝑅
3. ∆𝑄𝑇𝑅 ≅ ∆𝑆𝑅𝑇
1. ∠𝐻𝐼𝐽 ≅ ∠𝐾𝐼𝐽, ∠𝐼𝐽𝐻 ≅ ∠𝐼𝐽𝐾
2. 𝐽�̅� ≅ 𝐽�̅� 3. ∆𝐻𝐼𝐽 ≅ ∆𝐾𝐼𝐽
Given ASA Triangle Congruence SSS Triangle Congruence AAS Triangle Congruence SAS Triangle Congruence HL Triangle Congruence
Given ASA Triangle Congruence SSS Triangle Congruence AAS Triangle Congruence SAS Triangle Congruence HL Triangle Congruence
Given ASA Triangle Congruence SSS Triangle Congruence AAS Triangle Congruence SAS Triangle Congruence HL Triangle Congruence
Geometry-Congruent Triangles ~11~ NJCTL.org
Congruent Triangle Proofs – Honors
Classwork
PARCC-type problems
Write a two-column proof.
68. Given: ∠K ≅ ∠L, KL ≅ LM
Prove: ∆JKL ≅ ∆PML
69. Given: LOM NPM,
LM NM
Prove: ∆LOM ∆NPM
67. Given: ,BC DC AC EC
Prove: ABC ≅ EDC
Statements Reasons
Geometry-Congruent Triangles ~12~ NJCTL.org
Congruent Triangle Proofs – Honors
Homework
PARCC-type problems
Write a two-column proof.
71.
72. Given: HIJ KIJ
IJH IJK
Prove: ∆HIJ ∆KIJ
70. Given: || , WX YZ WX YZ
Prove: WXZ YZX
Geometry-Congruent Triangles ~13~ 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.
73. 74. Given: HI JG
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 are .
4) 4) Reflexive Prop. of
5) ∆FGK ∆HGK 5)
6) 6) CPCTC
Statements/Reasons Bank: 𝐾𝐹 ≅ 𝐾𝐻 CPCTC
𝐺𝐾 ≅ 𝐺𝐾 Vertical angles are congruent ∠𝐺𝐾𝐹 & ∠𝐺𝐾𝐻 are right ∡𝑠 Given
AAS Triangle Congruence ∠𝐴𝐷𝐵 & ∠𝐶𝐷𝐵 are right angles HL Triangle Congruence Transitive property of ≅
Geometry-Congruent Triangles ~17~ 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.
73. 74. Given: HI JG
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 are .
4) 4) Reflexive Prop. of
5) ∆FGK ∆HGK 5)
6) 6) CPCTC
Geometry-Congruent Triangles ~18~ NJCTL.org
76. Write a proof.
Given: YA BA , B Y
Prove: AZ AC
Statements Reasons
Geometry-Congruent Triangles ~19~ 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) ABCE is a rectangle. D is
the midpoint of CE .
1) Given
2) AED BCD 2) Definition of rectangle
3) AE BC 3) Definition of rectangle
4) 4)
5) 5)
6) 6)
Geometry-Congruent Triangles ~20~ NJCTL.org
80. Write a proof.
Given: BD AC , D is the midpoint of AC .
Prove: BC BA
Statements Reasons
Geometry-Congruent Triangles ~21~ 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. 94.
Geometry-Congruent Triangles ~22~ 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. 102. 103. 104. 105.
106. 107. 108.
Geometry-Congruent Triangles ~23~ 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°
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
b. ASA
c. AAS
d. Not congruent
7. By which postulate/theorem, if any, are the two triangles congruent?
a. ASA c. SAS
b. AAS d. Not congruent
R P
Q S
V U
y°5x
32°32°
Geometry-Congruent Triangles ~24~ NJCTL.org
8. State the third congruence needed to make ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 true by SAS congruence.
Given: a. 𝐴𝐶 ≅ 𝐷𝐹
∠B ≅ ∠E b. 𝐵𝐶 ≅ 𝐸𝐹
𝐴𝐵 ≅ 𝐷𝐸 c. ∠C ≅ ∠F d. ∠A ≅ ∠D
9. What information must be true for ASA congruence between the two triangles?
a. 𝐻𝐼 ≅ 𝐾𝐿
b. 𝐺𝐻 ≅ 𝐽𝐾
c. ∠I ≅ ∠L
d. 𝐺𝐼 ≅ 𝐽�̅�
10. State the third congruence needed to make ∆𝑋𝑌𝑍 ≅ ∆𝑃𝑄𝑅 true by ASA congruence.
Given: a. XY ≅ PQ ∠P ≅ ∠X b. PQ ≅ 𝑌𝑍 ∠Y ≅ ∠Q c. ∠X ≅ ∠P
d. 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. 12.
13. Find the value of each variable in the figure below.
7z°
(12y + 2)°(2x)°
(x + 5)°
A B
C D
X
A
M
X
I
N
95°
28°28°
95°
H
G
I L
J
K
Geometry-Congruent Triangles ~25~ 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
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 Reasons
1.) 𝐻𝐼 ⊥ 𝐺𝐽 1.)
2.) ∠𝐻𝐼𝐺 & ∠𝐻𝐼𝐽 are right angles 2.) 3.) ∠𝐻𝐼𝐺 ≅ ∠𝐻𝐼𝐽 3.)
4.) 𝐺𝐻 ≅ 𝐽𝐻 4.)
5.) 𝐻𝐼 ≅ 𝐻𝐼 5.) 6.) ∆HIG ≅ ∆HIJ 6.)
7.) 𝐺𝐼 ≅ 𝐽𝐼 7.)
8.) I is the midpoint of 𝐺𝐽 8.)
Honors:
15. Write a two-column or flow proof.
Given: 𝑀𝑁 ≅ 𝑀𝑋 , ∠𝐼 ≅ ∠𝐴
Prove: 𝑁𝐼 ≅ 𝑋𝐴
A
M
X
I
N
IG J
HReasons Bank
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