This document discusses different theorems for proving triangles are congruent: - 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

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. 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.

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.
SSS
SSS
ASA
AAS
HL
SAS
AAS
not ≅
not ≅

10. Examples: What additional information is required in order
to know that the triangles are congruent by the given
theorem?
1. AAS 2. HL
3. SAS 4. ASA

11. Examples: What additional information is required in order
to know that the triangles are congruent by the given
theorem?
1. AAS 2. HL
3. SAS 4. ASA
orGI JL IH LK≅ ≅
orAC LN CB NM≅ ≅
∠KLJ ≅ ∠DJL
∠NLW ≅ ∠XVW