36
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

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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

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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

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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

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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.

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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.

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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.

47. ASA Triangle 48. ASA Triangle 49. ASA Triangle

A X ZY CB AC XZ

AB XY Y B C Z For numbers 50 – 56, if the triangles are congruent, state which theorem applies and write the congruence statement. 50. 51. 52.

53. 54. 55.

56.

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Geometry-Congruent Triangles ~8~ 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 59. ASA Triangle

K O LP UY U L

PK YO Y P 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.

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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

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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

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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

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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

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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

SSS Triangle Congruence 𝐹𝐺 ≅ 𝐻𝐺 SAS Triangle Congruence ∠𝐹 ≅ ∠𝐻

ASA Triangle Congruence ∠𝐹𝐺𝐾 ≅ ∠𝐻𝐺𝐾 AAS Triangle Congruence ∠𝐺𝐾𝐹 ≅ ∠𝐺𝐾𝐻

HL Triangle Congruence Transitive property of ≅

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Geometry-Congruent Triangles ~14~ 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 CPCTC

𝑌𝑍 ≅ 𝐵𝐶 Vertical angles are congruent Given Reflexive property of ≅

SSS Triangle Congruence 𝑌𝐴 ≅ 𝐵𝐴 SAS Triangle Congruence ∠𝑍 ≅ ∠𝐶

ASA Triangle Congruence ∠𝐵 ≅ ∠𝑌 AAS Triangle Congruence ∠𝑍𝐴𝑌 ≅ ∠𝐶𝐴𝐵

HL Triangle Congruence Transitive property of ≅

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Geometry-Congruent Triangles ~15~ 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 Reasons

1) ABCE is a rectangle. D is

the midpoint of CE .

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 CPCTC

𝐴𝐷 ≅ 𝐵𝐷 Vertical angles are congruent Given Reflexive Property of ≅

SSS Triangle Congruence 𝐷𝐸 ≅ 𝐷𝐶 SAS Triangle Congruence ∠𝐸𝐴𝐷 ≅ ∠𝐶𝐵𝐷

ASA Triangle Congruence ∠𝐴𝐷𝐸 ≅ ∠𝐵𝐷𝐶 AAS Triangle Congruence ∠𝐴𝐷𝐵 ≅ ∠𝐴𝐷𝐵 HL Triangle Congruence Definition of Midpoint

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Geometry-Congruent Triangles ~16~ 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 Vertical angles are congruent

𝐶𝐴 ≅ 𝐴𝐶 All right angles are congruent Given Reflexive property of ≅

SSS Triangle Congruence 𝐵𝐶 ≅ 𝐵𝐴 SAS Triangle Congruence ∠𝐴 ≅ ∠𝐶 ASA Triangle Congruence ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷

AAS Triangle Congruence ∠𝐴𝐷𝐵 & ∠𝐶𝐷𝐵 are right angles HL Triangle Congruence Transitive property of ≅

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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

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Geometry-Congruent Triangles ~18~ NJCTL.org

76. Write a proof.

Given: YA BA , B Y

Prove: AZ AC

Statements Reasons

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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)

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Geometry-Congruent Triangles ~20~ NJCTL.org

80. Write a proof.

Given: BD AC , D is the midpoint of AC .

Prove: BC BA

Statements Reasons

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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.

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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.

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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°

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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

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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

triangle are ≅

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Unit 5 - Congruent Triangles - ANSWER KEY Congruent Triangles Classwork

1. a. ) XY b.) CB c.) YXZ d.) AC e.) C f.) Y

2. a.) CA JS , AT SD, CT JD

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

Angles: R V, T W, S U

Congruence Statement:

RTS

VWU

5. Sides: CA XZ ,BC YX, AB ZY (Since the triangles are isosceles, other answers may be correct.)

Angles: B Y, C 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.

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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

Angles: W Y, Q 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 c.) Vertical Angles are congruent

d.) Third Angle Theorem e.) All corresponding parts are congruent, so triangles are congruent.

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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

27. SAS Triangle Congruence;

MKL

NPO

28. Not enough information

29. SAS Triangle Congruence;

WZV

XZY

30. SSS Triangle Congruence;

ABD

CDB

31. Not enough information

32. SAS Triangle Congruence;

JMK

LMK

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)

40. SAS Triangle Congruence;

RSE

PTJ

41. Not enough information

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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

46. SSS/SAS Triangle Congruence;

JIL

LKJ

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Proving Congruence (Triangle Congruence: ASA, AAS and HL) Classwork 47. B Y

48. A X

49. A X

50. ASA Triangle Congruence;

TUV

WXY 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)

Homework

57. P Y

58. K O

59. KL OU

60. AAS Triangle Congruence;

BCE

DCF 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

B

AC Z

X

Y

#48

A

B

C X

Y

Z

#49

A

B

C X

Y

Z

O

U

YP

L

K

#58

K

L

P Y

U

O

#59

K

L

P Y

U

O

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Congruent Triangle Proofs – both CP & Honors have the same answers

Classwork

67. Statements Reasons____________

1. BC DC; AC EC 1. Given

2. BCA DCE 2. Vertical Angles are congruent

3.

ABC

EDC 3. SAS Triangle Congruence

68. Statements Reasons____________

1. K M ; KL ML 1. Given

2. JLK PLM 2. Vertical Angles are congruent

3.

JKL

PML 3. ASA Triangle Congruence

69. Statements 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

71. Statements Reasons____________

1. Q S ; TRS TRQ 1. Given

2. RT TR 2. Reflexive Property of Congruence

3.

QTR

SRT 3. AAS Triangle Congruence

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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

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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

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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

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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 Reasons

1.) 𝐻𝐼 ⊥ 𝐺𝐽 1.) Given

2.) ∠𝐻𝐼𝐺 & ∠𝐻𝐼𝐽 are right angles 2.) Def. of perpendicular lines

3.) ∠𝐻𝐼𝐺 ≅ ∠𝐻𝐼𝐽 3.) All right angles are congruent

4.) 𝐺𝐻 ≅ 𝐽𝐻 4.) Given

5.) 𝐻𝐼 ≅ 𝐻𝐼 5.) Reflexive Property of Congruence 6.) ∆HIG ≅ ∆HIJ 6.) HL

7.) 𝐺𝐼 ≅ 𝐽𝐼 7.) CPCTC

8.) I is the midpoint of 𝐺𝐽 8.) Definition of Midpoint

15. Statement Reason

𝑀𝑋 ≅ 𝑀𝑁 given

∠I ≅ ∠A given ∠NMI ≅ ∠XMA vertical angles are congruent

∆NMI ≅ ∆XMA AAS

𝑁𝐼 ≅ 𝑋𝐴 CPCTC

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