3.2 Three Ways To Prove
Triangles Congruent
Objective:
After studying this lesson you will be able to
identify included angles and included sides as
well as apply the SSS, SAS, ASA postulates.
In the figure at the right, H is
included by the sides GH and HJ.
G
H
J
Which sides include
G?
Which angles include HJ?
Triangles have some special properties to help
us prove that 2 triangles are congruent using
only 3 specially chosen pairs of corresponding
parts.
If you have three toothpicks (sides) of different
lengths you can create a triangle. If your friend
has three toothpicks (sides) that have the same
measure as your toothpicks then your friend can
create a triangle that is congruent to the one
that you built.
B
T
W
O
E
Postulate If there exists a correspondence
between the vertices of two triangles
such that three sides of one triangle
are congruent to the corresponding
sides of the other triangle, the two
triangles are congruent. (SSS)
I
B
T
W
O
E
Postulate If there exists a correspondence
between the vertices of two triangles
such that two sides and the
included angle of one triangle are
congruent to the corresponding
parts of the other triangle, the two
triangles are congruent. (SAS)
I
B
T
O
W
E
Postulate If there exists a correspondence
between the vertices of two triangles
such that two angles and the
included side of one triangle are
congruent to the corresponding
parts of the other triangle, the two
triangles are congruent. (ASA)
I
T
Using the congruent markings
determine what is needed to
prove the triangles congruent
using the specific methods.
T
O
M
SSS
SAS
A
C
T
D
A
O
SAS
ASA
N
M
P
R
W
V
T
S
Prove PWT  SVR
SAS
ASA
Given: AD  CD
B is the midpoint of AC
Prove: ABD  CBD
A
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
D
B
C
Given: 3  6
KR  PR
KRO  PRM
Prove: KRM  PRO
Statement
1. 3  6
3 is supp. 4
3. 5 is supp. 6
4.  4  5
2.
R
3 4
K
KR  PR
6. KRO  PRM
7. KRM  PRO
8.
KRM  PRO
O
Reason
1. Given
2. If 2 angles form a straight angle, they are supp.
3. If 2 angles form a straight angle, they are supp.
4. angles supp. to congruent angles are
congruent
5.
M
5 6
P
5.
Given
6.
Given
7.
Subtraction property
8.
ASA (steps 4, 5, 7)
Summary:
How many parts are there
in proving triangles
congruent? What are the
shortcuts that we can use
to prove triangles
congruent?
Homework: worksheet