Geometry-Triangle Congruency
Review Sheet
Name:_____________
Provide complete solutions to each of the following problems. Diagrams are provided to
illustrate general shapes and clarify the location of labels. Draw conclusions based on “given”
information, not based on the apparent shape of a diagram.
1.
For each part of this problem, you are given some information. Determine whether
this information is sufficient to justify the conclusion. Justify your answer.
E
a.
Suppose 1 = 2 and DF = GC. Is this enough
information to conclude that ∆DEF  ∆CEG?
1
D
2
C
G
F
A
B
E
b. Suppose ∆ADF  ∆BCG and 3 = 4. Is this
enough information to conclude that ∆DFE  ∆CGE?
D
F
3
A
2.
C
G
4
B
Given: C is the midpoint of BD; C is also the midpoint of AE.
Prove that BAC = CED.
A
C
D
B
E
3.
Complete the proof below by providing an appropriate reason for each statement.
A
Given:
ABC, with AB = AC,
BQ bisects ABC,
CP bisects ACB, and
BQ meets CP at X
P
Prove:
PX = QX
X
B
1. AB = AC
2. ABC= ACB
3. ½ABC= ½ACB
4. XBC = ½ABC
XCB = ½ACB
5. XBC = XCB
6. BX = CX
7. PBX = ½ABC
QCX = ½ACB
8. PBX = QCX
9. PXB = QXC
10.  PXB   QXC
11. PX = QX
Q
C
4.
If you know that three corresponding parts of two different triangles are equal, then
you may or may not be able to conclude that the triangles are congruent. For each of
the six cases below, if it is sufficient to prove congruence, then circle the three letter
abbreviation; if it is not sufficient to prove congruence, then sketch two triangles
with matching parts that are clearly not congruent.
a. SSS (three corresponding sides are equal)
b. SAS (two corresponding sides and an included angle are equal)
c.
AAA (three corresponding angles are equal)
d. ASA (two corresponding angles and an included side are equal)
e. SSA (two corresponding sides and a non-included angle are equal) .
f. AAS (two corresponding angles and a non-included side are equal)
5.
Given the diagram at right with CB = CD, BA = DE.
Prove BDF is isosceles.
A
E
B
F
D
C
6. Definition: A perpendicular bisector is a ray that bisects a segment and is
perpendicular to the segment.
a. Prove the statement: If ABC is isosceles with AB = AC, then the angle bisector
of A is the perpendicular bisector of BC.
b. Write the converse to the statement in part a. Is the converse also true? Justify
your reasoning.
7.
Tell which pairs of congruent/equal parts (in the
figure at right) and what triangle congruence method
you would use to prove the triangles in parts a. and b.
congruent.
B
Each part should be answered independently from the
other. In other words, the information in part a.
should not be used for part b.
A
a.
3
F 5
6
G
4
If 1  2, 3  4 and AE  CD, why is
∆ABE  ∆CBD?
1
2
E
D
b. If 3  4 and 5  6, and AF  GC why is ∆ABG  ∆CBF?
8. Given: I is the midpoint of segment HJ, M is the midpoint of segment HL, HJ =
HL,
J
and NJK = NLK.
Determine which triangle is congruent to ∆NHJ.
Write sentences explaining your reasoning.
I
K
N
H
M
L
C
9.
Complete the proof below by providing an appropriate reason for each statement.
Given: AB = CD, AD = BC,
AX  AD, CY BC, and
BXYD is a line segment
A
Prove: AX = CY
B
Y
X
C
Statements
1) AB = CD
Reasons
1)
2) AD = BC
2)
3) AX  AD and CY BC
3)
4) BD = BD
4)
5) ABD  CDB
5)
6) CBD = ADB
6)
7) BCY= DAX = 90
7)
8) BCY  DAX
8)
9) AX = CY
9)
D
J
10. GIVEN: M is a midpoint. RM=9x+4, MI=5x+20,
RJ=15x+20, JI=12x+82
Is  JRI an isosceles triangle? Support your answer.
R
M
11. Prove the following.
D
GIVEN: ABCDE is a pentagon (a five-sided
polygon) with all sides and all angles equal.
C
PROVE: CEA CAE
E
B
A
12) Complete the following proof:
B
Given: AE bisects
AB  AD
BAD
A
Prove:
E
C
BCE  ECD
D
I
13) For each of the following, either state that the triangles can be found congruent (and
state which triangle congruency you would use), or state that they can’t be found
congruent and draw a counter example.
a)
Given:
A E
B D
D
A
C
E
ABC  ECD
B
Counter-example (if necessary):
b)
Given: C bisects BE and AD
B
D
C
ACB  DCE
A
E
Counter-example (if necessary):
c)
Given:
B
BCA  DCE
BC  CD
AB  DE
A
ABC  EDC
Counter-example (if necessary):
D
C
E
d)
Given: AC bisects
B
BAD and
BCD
C
A
ACB  ACD
D
Counter-example (if necessary):
14) Given the following statement: All equilateral triangles are isosceles triangles.
a) Write the statement in “if…, then…” form:
b) Write the converse of the statement:
c) Is the converse true? If not, justify your answer.
15) Given the following conditional statement, set-up a proof by providing a diagram, the
given information, and the prove statement (only set-up the proof):
If a point on the base of an isosceles triangle is equidistant from the
midpoints of the legs, then that point is the midpoint of the base.
Diagram
Given:
Prove:
16) Complete the following proof:
Given : SV bisects
Prove:
1
S,
SRT  RTS
S
2
V
R
Statements
2
1
T
Reasons
1) SV bisects S
1)_____________________________
2) _____________________
2)_____________________________
3)
SRT  RTS
4) ST  RS
5) ____________________________
6) RSV  TSV
7) VT  RV
8)____________________________
3) Given
4)_____________________________
5) Reflexive Property
6) ____________________________
7)_____________________________
8)_____________________________
17) Complete the following proof:
Given: AB  BC  CD  DA
C
D
Prove: AC  BD
B
A
18) Complete the following proof:
Given: AB  AC
RB bisects
ABP
TC bisects
ACQ
S
P
C
B
Q
Prove: SB  SC
A
R
T
19) Complete the following proof:
Given: T is the midpt. of MN
PMT and QNT are
right angles
MR  SN
1 2
Prove:
Q
P
Q
P
S
R
2
1
M
T
N