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Congruent Triangles
Triangle Congruence Theorems
Five valid methods for proving that triangles are congruent are given below.
SAS
SSS
E
B
A
D
E
A
Two sides and the
included angle
are congruent.
a.
B
F
C
Example 1
HL (right triangles only)
D
ASA
E
B
F
C
All three sides
are congruent.
E
D
A
AAS
B
F
C
A
The hypotenuse and one
of the legs are congruent.
D
E
B
F
C
A
Two angles and
the included side
are congruent.
D
F
C
Two angles and a
non-included side
are congruent.
Determine whether there is enough information to prove that the triangles are congruent.
Explain your reasoning.
A
b.
B
K
C
E
D
J
M
L
— ≅ EC
—. By the Vertical Angles Congruence Theorem,
a. You are given that ∠ A ≅ ∠ E and AC
∠ ACB ≅ ∠ ECD. So, two pairs of angles and their included sides are congruent. By the
ASA Congruence Theorem, △ABC ≅ △EDC.
— ≅ LK
—. You know that ∠ J ≅ ∠ L by the Base Angles Theorem. You also
b. You are given that JK
—
—
know that KM ≅ KM by the Reflexive Property of Segment Congruence. Because two pairs of
sides and their non-included angles are congruent, you cannot conclude that △JKM ≅ △LKM.
Practice
Check your answers at BigIdeasMath.com.
Determine whether there is enough information to prove that the triangles are
congruent. If so, state the theorem you would use.
1.
X
2.
Y
W
E
F
3. no
R
D
G
Q
T
Z
yes; SSS Congruence Theorem
S
yes; HL Congruence Theorem
4.
no
C
5.
G
H
B
6.
U
S
D
A
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F
I
yes; AAS Congruence Theorem
T
V
R
yes; ASA Congruence Theorem
Topic 11.5