Congruent triangles have congruent sides and congruent angles.
Unit 6 - Congruent Triangles Congruent Triangles...
Transcript of Unit 6 - Congruent Triangles Congruent Triangles...
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.
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.
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.
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
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 ≅
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 ≅
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
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 ≅
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
triangle are ≅
Geometry-Congruent Triangles ~26~ NJCTL.org
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.
Geometry-Congruent Triangles ~27~ 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
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.
Geometry-Congruent Triangles ~28~ 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
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
Geometry-Congruent Triangles ~29~ 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
46. SSS/SAS Triangle Congruence;
JIL
LKJ
Geometry-Congruent Triangles ~30~ NJCTL.org
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
Geometry-Congruent Triangles ~31~ NJCTL.org
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
Geometry-Congruent Triangles ~32~ 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 ~33~ 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 ~34~ 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 ~35~ 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 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
Geometry-Congruent Triangles ~36~ NJCTL.org