This document discusses different congruence theorems for triangles: - Side-Side-Side (SSS) - If three sides of one triangle are congruent to three sides of another, the triangles are congruent. - Side-Angle-Side (SAS) - If two sides and the included angle of one triangle are congruent to those of another, the triangles are congruent. - Angle-Side-Angle (ASA) - If two angles and the included side of one triangle are congruent to those of another, the triangles are congruent. - Angle-Angle-Side (AAS) - If two angles and a non-included side of one triangle are congruent to those

1. Congruent Triangles
The student is able to (I can):
• Identify and prove congruent triangles given
– Three pairs of congruent sides (Side-Side-Side)
– Two pairs of congruent sides and a pair of congruent
included angles (Side-Angle-Side)
– Two angles and the included side (Angle-Side-Angle)
– Two angles and the non-included corresponding angle
(Angle-Angle-Side)
– A Hypotenuse and a Leg of a right triangle

2. SSS – Side-Side-Side
If three sides of one triangle are congruent to three sides of
another triangle, then the triangles are congruent.
T
I
N
C
U
P
4
6
7 4
6
7
ΔTIN ≅ ΔCUP

3. SAS – Side-Angle-Side
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of another
triangle, then the triangles are congruent.
L
H
S
U
T
A
ΔLHS ≅ ΔUTA

4. ASA – Angle-Side-Angle
If two angles and the included side of one triangle are
congruent to two angles and the included side of another
triangle, then the triangles are congruent.
F
L
Y
B U
G
ΔFLY ≅ ΔBUG

5. AAS – angle-angle-side
If two angles and a nonnonnonnon----includedincludedincludedincluded side of one triangle are
congruent to two angles and a non-included correspondingcorrespondingcorrespondingcorresponding
side of another triangle, then the triangles are congruent.
The non-included sides mustmustmustmust be corresponding in
order for the triangles to be congruent.
N
I
W
UO
Y
ΔYOU ≅ ΔWIN

6. HL – hypotenuse-leg
If the hypotenuse and leg of one right triangle are congruent
to the hypotenuse and leg of another right triangle, then the
two triangles are congruent.
J
O
E
M
AC
ΔJOE ≅ ΔMAC

7. ASS – angle-side-side
(We do not cuss in Geometry)
There is no ASS (or SSA) congruence theorem.
(unless the angle is a right angle – then it would be HL)

8. The diagrams and triangles you will be examining will often
assume that you will recognize congruent pieces that may
not be marked. These include:
• Shared side/angle – if a triangle shares a side or an angle
with another triangle, the shared side/angle is congruent
to itself
• Vertical angles –all vertical angles are congruent
• Parallel lines – corresponding, alternate interior, and
alternate exterior angles of parallel lines are congruent
• Midpoint/bisector – the midpoint of a segment (or
bisector of an angle) cuts the segment (or angle) into two
congruent pieces.
These are the ONLY markings you should add to a diagram.These are the ONLY markings you should add to a diagram.These are the ONLY markings you should add to a diagram.These are the ONLY markings you should add to a diagram.
Do NOT add any markings because you “think” they lookDo NOT add any markings because you “think” they lookDo NOT add any markings because you “think” they lookDo NOT add any markings because you “think” they look
congruent.congruent.congruent.congruent.

9. Examples: Decide if SSS, SAS, ASA, AAS, or HL would prove
the triangles congruent. If the triangles are not congruent,
use “Not ≅”.
1. 2. 3.
4. 5. 6.
7. 8. 9.

10. Examples: Decide if SSS, SAS, ASA, AAS, or HL would prove
the triangles congruent. If the triangles are not congruent,
use “Not ≅”.
1. 2. 3.
4. 5. 6.
7. 8. 9.
SSS
SSS
ASA
AAS
HL
SAS
AAS
not ≅
not ≅