Triangle Proofs
____ 1. Justify the last two steps of the proof.
Given:
and
Prove:
R
S
T
U
Proof:
1.
2.
3.
4.
1. Given
2. Given
3.
4.
A. Symmetric Property of ; SSS
C. Reflexive Property of ; SSS
B. Reflexive Property of ; SAS
D. Symmetric Property of ; SAS
____ 2. What other information do you need in order to prove the triangles congruent using
the SAS Congruence Postulate?
A
B
C
A.
B.
D
C.

D.

____ 3. State whether
and
7
A.
B.
C.
D.
are congruent. Justify your answer.
7
yes, by either SSS or SAS
yes, by SSS only
yes, by SAS only
No; there is not enough information to conclude that the triangles are
congruent.
____ 4. Which triangles are congruent by ASA?
F
A
(
V
T
((
G
(
)
((
H
U
C
A.
C.
B.
D. none
B
____ 5. Which two triangles are congruent by ASA?
N
M
O
P
Q
A.
B.
R
C.
D. none
____ 6. Which pair of triangles is congruent by ASA?
A.
C.
B.
D.
____ 7. What is the missing reason in the two-column proof?
Given:
bisects
and
bisects
Prove:
N
M
<
O
>
P
Statements
1.
bisects
2.
3.
4.
5.
Reasons
1. Given
2. Definition of angle
bisector
3. Reflexive property
4. Given
5. Definition of angle
bisector
6.
?
bisects
6.
A. ASA Postulate
C. AAS Theorem
B. SSS Postulate
D. SAS Postulate
A
(
____ 8. Name the theorem or postulate that lets you immediately conclude
B
D
(
C
A. AAS
B. SAS
C. ASA
D. none of these
____ 9. If
and
conclude that
, which additional statement does NOT allow you to
?
A.
B.
C.
D.
____10. Can you use the SAS Postulate, the AAS Theorem, or both to prove the triangles
congruent?
A. either SAS or AAS
B. SAS only
C. AAS only
D. neither
____11. What else must you know to prove the triangles congruent by ASA? By SAS?
B
(
A
(
D
A.
B.
C
;
;
C.
D.
;
;
____12. Based on the given information, what can you conclude, and why?
Given:
I
K
J
H
A.
B.
L
by ASA
by SAS
C.
D.
____13. From the information in the diagram, can you prove
A. yes, by ASA
B. yes, by AAA
? Explain.
C. yes, by SAS
D. no
____14. For which situation could you immediately prove
A. I only
by ASA
by SAS
B. II only
C. III only
using the HL Theorem?
D. II and III
____15. Is there enough information to conclude that the two triangles are congruent? If so,
what is a correct congruence statement?
A
|
|
B
A.
B.
C.
D.
C
D
Yes;
.
Yes;
.
Yes;
.
No, the triangles cannot be proven congruent.
____16. What additional information will allow you to prove the triangles congruent by the HL
Theorem?
A
B
|
C
D
A.
B.
|
E
C.
D.
17. Are the triangles congruent? Justify your answer.
18. Are
congruent? Justify your answer.
A
5
16
16
5
D
B
C
19. Based on the given information, can you conclude that
Given:
,
, and
? Explain.
20. Is there enough information to prove the two triangles congruent? If yes, write the
congruence statement and name the postulate you would use. If no, write not possible
and tell what other information you would need.
Q
|
|
(
)
S
P
R
21. Write the missing reasons to complete the proof.
Given:
,
, and
Prove:
E
B
A
D
C
F
Statement
1.
2.
3.
Reason
1. Given
2. Given
3. Given
4. Definition of congruent
segments
5.
?
6. Segment Addition Postulate
7. Definition of congruent
segments
8.
?
4.
5.
6.
7.
8.
22. Write a two-column proof to show that
Given:
and
N
P
Q
O
M
R
23. Is
by HL? If so, name the legs that allow the use of HL.
A
B
D
C
24. Complete the proof by providing the missing reasons.
Given:
Prove:
S
H
Statement
1.
2.
3.
4.
?
5.
D
T
Reason
1. Given
2.
?
3.
?
4. Reflexive Property
5.
?
25. Write the missing reasons to complete the flow proof.
Given:
Prove:
are right angles,
B
(
)
A
C
D
26. Name a pair of triangles in the figure and state whether they are congruent by SSS,
SAS, ASA, AAS, or HL.
Given:
,
N
O
M
P
27. Complete the proof by providing the missing reasons.
Given:
Prove:
,
C
D
B
E
A
Statement
1.
,
2.
angles
3.
4.
5.
Reason
, and
are right
28. Write a proof to show that
Given:
and
B
D
C
A
E
1. Given
2. Definition of perpendicular
segments
3.
?
4.
?
5.
?
.
Triangle Proofs
Answer Section
1. C
9. B
2. B
10. C
3. A
11. B
4. B
12. A
5. B
13. A
6. D
14. C
7. A
15. B
8. A
16. C
17. Yes, the diagonal segment is congruent to itself, so the triangles are congruent by
SAS.
18. Yes, the diagonal segment,
is congruent to itself.
, so
.
, so
. The triangles are congruent by the definition of congruent
triangles.
19. Answers may vary. Sample: Two pairs of sides are congruent, but the angle is not
included. There is no SSA Congruence Theorem, so you cannot conclude
with the information given.
20. Yes;
by SAS.
21. Step 5: Addition property of equality
Step 8: SAS
22.
23. Yes,
(in each triangle)
24. 2. Definition of
3. Given
4.
5. HL Theorem
lines
25. a. Definition of right triangles
b. Converse of Isosceles Triangle Theorem
c. Reflexive property
d. HL Theorem
26.
by SSS
27. Step 3: All right angles are congruent.
Step 4: Reflexive Property
Step 5. HL Theorem
28.
Answers may vary. Sample: You are given that
. Vertical angles BCA and ECD are congruent, so
SAS.
and
by