Geometry
Name______________________
Proving Congruent Triangles
Date: Sept. 23, 2011
Congruent Triangles
Since triangles are simply enclosed figures made up of only three sides and three angles, then
the following definition should follow:
Definition: If each of three sides of a triangle is congruent to the corresponding sides of
another triangle and the corresponding angles are also congruent, then the triangles are
congruent.
Suppose, for example, we are given two triangles (such as the ones below):
W
B
C
U
V
A
Specifically, we know ABC  WVU if all corresponding angles and corresponding sides are
congruent.
To prove that all six bits are congruent every time we want to prove two triangles are
congruent would be a real pain. We therefore need to understand that there are certain
MINIMUM conditions which guarantee congruence. We will study these minimum conditions
in their entirety, but we will begin by looking at two of these scenarios, each stated as a
postulate:
Side-Angle-Side (SAS): -- Postulate
If two sides of one triangle are congruent to two sides of a second triangle AND the included
angles are also congruent, then the triangles are congruent. Place tick marks for this scenario.
C
D
A
B
G
H
Geometry
Angle-Side-Angle (ASA): -- Postulate
If two angles of one triangle are equal to two angles of another triangle AND the included
sides are also congruent, then the triangles are congruent. Place tick marks for this scenario.
C
R
T
A
S
B
Extension from Congruent Triangles
Suppose we look at a scenario where we have proven ABC  WVU , using SAS, where
AB  WV , B  V , and A  U . Make tick markings on the triangles below.
W
B
C
U
V
A
Q: What conclusion can we draw concerning all remaining corresponding parts of these
congruent triangles?
We can justify this using the acronym:
Q: What do you think CPCTC stands for?
C
P
C
T
C
Geometry
Exercises:
Directions: Perform the following proof. Note that you will end up proving that two triangles
are congruent. Use your prior and newly gained knowledge to complete the following proofs.
1)
Q
Given: AQ  BC
1  2
AP  AR
Prove:
P
1) APQ  ARQ
2) AQP  AQR
1
B
Statements
R
2
A
Reasons
C
Geometry
2)
O
Given: OP  MR
3  4
M  R
P is the midpt. of MR
N
Prove: N  Q
Q
4
3
M
Statements
P
Reasons
R