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PART II REVIEW – TRIANGLES & CONGRUENCE
Are the triangles congruent? If yes, justify with a congruence theorem and write the congruence statement.
Otherwise, write “no” (if they are definitely not congruent) or “not enough information” if you don’t have enough
information. You can add information that is not marked if you know it is true.
Yes/No? _________________
Yes/No? _________________
Yes/No? _________________
Statement: ______________
Statement: ______________
Statement: ______________
Yes/No? _________________
Yes/No? _________________
Yes/No? _________________
Statement: ______________
Statement: ______________
Statement: ______________
Yes/No? _________________
Yes/No? _________________
Yes/No? _________________
Statement: ______________
Statement: ______________
Statement: ______________
12.
Yes/No? _________________
Yes/No? _________________
Yes/No? _________________
Statement: ______________
Statement: ______________
Statement: ______________
Multiple Choice. Select the one best answer for each question.
13. In the diagram of ∆KLM, mÐL = 70, mÐM = 50 , and MK
is extended through N. What is the measure of ÐLKN ?
A) 60°
B) 120°
C) 180°
D) 300°
14. Lines s and t intersect as shown. Which angles are vertical?
A) Ð9 and Ð10
B) Ð7 and Ð9
C) Ð8 and Ð10
D) Ð7 and Ð10
15. The diagram at the right shows a pair of congruent triangles, with ÐADB @ ÐCDB and
ÐABD @ ÐCBD.
Which statement must be true?
A) ÐADB @ ÐCBD
B) ÐABC @ ÐADC
C) AB @ CD
D) AD @ CD
16. If ∆ ABC @ ∆ JKL @ ∆ RST , then AC must be congruent to
A) JL
B) JK
C) ST
D) RS
17. In the diagram below, four pairs of triangles are shown. Congruent corresponding parts are labeled
in each pair. Using only the information given in the diagrams, which pair of triangles can not be
proven congruent?
A)
B)
C)
D)
18. In the diagram of ∆ABC, AB is extended to point D. If mÐCAB = x + 40 , mÐACB = 3x +10 ,
mÐCBD = 6x , what is mÐCAB?
A) 13
B) 25
C) 53
D) 65
19. In ∆ABC, mÐA = x , mÐB = 2x + 2 , and mÐC = 3x + 4 . What is the value of x?
A) 29
B) 31
C) 59
D) 61
20. In the diagram of ∆AGE and ∆OLD, ÐGAE @ ÐLOD and AE @ OD. To prove ∆ ABGE @ ∆ OLD
by SAS, what additional information is needed?
A) GE @ LD
B) AG @ OL
C) ÐAGE @ ÐOLD D) ÐAEG @ ÐODL
21. If the measure of an angle is represented by x, which expression represents the measure of its
complement?
A) 180 – x
B) 90 + x
C) 90 – x
D) 180
22. mÐANG = 35° . What is the measure of an angle supplementary to ÐANG ?
A) 35°
B) 55°
C) 325°
23. What does CPCTC mean?
A)
B)
C)
D)
Corresponding parts of Congruent Triangles are Corresponding
Corresponding parts of Congruent Triangles are Congruent
Congruent parts of Congruent Triangles are Congruent
Congruent parts of Corresponding Triangles are Congruent
D) 145°
24. Use appropriate symbols and letters to name these geometric figures.
a. _____________
b. _______________
G
c. _______________
D
E
F
D
B
C
25. Sketch the following:
a. acute angle ÐWMK
d. _______________
m
b. straight angle ÐBCD
c.
d. ⊙ O , with diameter AB
e. A pair of complementary
angles
f. A pair of supplementary
angles
g. an exterior angle
h. Vertical angles
i. An obtuse angle
26. AB and CD intersect at E. If mÐAEC = 5x - 30 and mBED  x  50, find, mCEB.
(Create a sketch to help you)
27. Solve for a.
28. Solve for x.
Triangles are drawn with certain measures indicated. Are all triangles with these measures congruent? Indicate
why/why not.
29.
30.
31.
1.6"
1"
23°
1.5"
78°
2.3"
Yes/No? ___________
Why/Why Not?
63°
Yes/No? ___________
Why/Why Not?
Yes/No? ___________
Why/Why Not?
Solve for x and/or y in each of the figures below. Be sure to show the set-ups for each problem.
32.
33.
x+6
7x + 10
x + 20
x = ________, y = _________
35. After you find x, find mÐA .
x = ________
34.
x = ____________ ,
mÐA = _______________
x = ________, y = _________
36. After you find x, find
x = ____________ ,
mÐA .
37. After you find x, find
mÐA = _______________
mÐA .
C
A
x+15°
2x+45°
B
x = ____________ ,
mÐA = ______________
38. Given two side lengths of a triangle, state what numbers the third side must be between.
a, 7, 10
between _________ and _________
b, 2.8, 13.5
between _________ and _________
c, 262, 145
between _________ and _________
39. Suppose the two triangles at the right are congruent.
a. Write a congruence statement for the two triangles. _________________
b. Name the angle included by the sides AC and BC . ________________
c. Name the side included between
ÐA and ÐC . ___________________
40. Based on the congruence statement and the diagram, fill in the missing measures.
∆ DEF @ ∆ DSR
mÐF = _____________
RD = _____________
mÐS = _____________
ED = _____________
mÐEDF = _____________
RS = _____________
mÐRDE = _____________
mÐRDS = _____________
41.
Given: AE bisects BD ,
ÐB @ ÐD
Prove: ∆ ABC @ ∆ DBC
Statements
Reasons
42.
Given: C is the midpoint of AE and BD
Prove: ∆ ABC @ ∆ EDC
Statements
43.
Given ÐL @ ÐN ,
Prove: ML @ ON
Reasons
ÐLOM @ ÐNMO
Statements
Reasons
44.
Given: LM bisects
Prove: JM @ KM
ÐJLK , LJ @ LK
Statements
45.
Reasons
Given: AC @ AD , CB @ DB
Prove:
ÐCAB @ ÐDAB
Statements
Reasons
ANSWERS – PART II
1. Not enough info. We need at least one pair of sides to be congruent.
2. Yes, AAS, DIJG @ DHGJ
3. Yes, AAS; DKOL @ DMNL
4. No, the sides do not correspond
5. Yes, ASA; DWVX @ DZUY
6. Yes; SAA, DACD @ DDBA
7. Yes, SSS or SAS; DFGE @ DIGH
8. Yes, H-L (hypotenuse-leg); DJLK @ DNOM
11. Yes, SAS; DCAD @ DCEB
9. Yes, SAS; DPQS @ DRSQ
10. not enough info
12. Yes, H-L theorem; DLMK @ DONK
13. B
14. D
24a. ED
15. D
16. A
24b. GF
24c.
25a.
17. D
19. A
ÐDBC or ÐCBD or ÐB
b.
f.
18. D
c.
g.
20. B
21. C
23. B
24d. DE or ED or m
d.
ÐKIJ
22. D
h.
e.
i.
26. 5x - 30 = x + 50 ® 4x = 80 ® x = 20 ® mÐCEB = 110°
27. 2a + 2(78) + 2(52) = 360 ® a = 50°
28. x + 90 + 4x = 160 ® 5x + 90 = 160 ® 5x = 70 ® x = 14
29. No. We only have two angles. We need at least on side to know all triangles with these measure are @
30. Yes. SSS
32.
33.
31. Yes. SAA
x + 6 + 7x +10 + x + 20 = 180 ® 9x + 36 = 180 ® x = 16
y +105 = 180 ® y = 75°
34.
x = 63°
35.
y = 54°
x + 60 = 105 ® x = 45°
3x - 6 + 8x + 4 = 130 ® 11x - 2 = 130 ® x = 12
36.
mÐA = 3(12) - 6 = 30°
37.
x + 37 + x + 67 = 90 ® 2x +114 = 90 ® x = -12
mÐA = -12 + 37 = 25°
x +15 + 2x + 45 = 90 ® 3x + 60 = 90 ® x = 10
mÐA = 10 +15 = 25°
38a. 3 and 17
38b. 10.7 and 16.3
39a. ∆ ABC @ ∆ A' B'C'
39b.
40. mÐF = 36°
RD = 15
mÐEDF = 92° RS = 21
ÐC
mÐS = 52°
mÐRDE = 88°
38c. 121 and 407
39c. AC
ED = 14
mÐRDS = 92°
41.
Statements
1. AE bisects BD
2. ÐB @ ÐD
3. ÐACB @ ÐECD
4. BC @ DC
5. ∆ ABC @ ∆ DBC
Reasons
1. Given
2. Given
3. Vertical angles are congruent
4. A bisector cuts a segment into 2 congruent segments
5. ASA
42.
Statements
1. C is the midpoint of AE and BD
2. BC @ DC , AC @ EC
3. ÐACB @ ÐECD
4. ∆ ABC @ ∆ EDC
Reasons
1. Given
2. Midpoints cut segments into 2 congruent segments
3. Vertical angles are congruent
4. SAS
43.
Statements
1. ÐL @ ÐN ,
ÐLOM @ ÐNMO
2. MO @ OM
3. ∆ LMO @ ∆ NOM
4. ML @ ON
Reasons
1. Given
2. Reflexive property of congruence
3. AAS
4. CPCTC
44.
Statements
1. LM bisects ÐJLK , LJ @ LK
2. ÐJLM @ ÐKLM
2. MO @ OM
4. ∆ JLM @ ∆ KLM
5. JM @ KM
Reasons
1. Given
2. An angle bisector cuts and angle into 2 congruent angles.
2. Reflexive property of congruence
4. SAS
5. CPCTC
45.
Statements
1. AC @ AD , CB @ DB
2. AB @ AB
3. ∆ ABC @ ∆ ABD
4. ÐCAB @ ÐDAB
Reasons
1. Given
2. Reflexive property of congruence
3. SSS
4. CPCTC