Determine congruent triangles

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 a side (Angle-Side-Angle and 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. Example Given: , D is the midpoint of
Prove: FRD ≅ ERD
F
R
ED
FR ER≅ FE
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. 1. Given
2. D is midpt of 2. Given
3. 3. Def. of midpoint
4. 4. Refl. prop. ≅
5. FRD ≅ ERD 5. SSS
FR ER≅
FE
FD ED≅
RD RD≅

4. 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

5. Example Given: , A is the midpoint of
Prove: FAR ≅ EAM F
R
A
M
E
FA EA≅ RM
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. 1. Given
2. ∠FAR ≅ ∠EAM 2. Vertical ∠s
3. A is midpt of 3. Given
4. 4. Def. of midpoint
5. FAR ≅ EAM 5. SAS
FA EA≅
RM
RA MA≅

6. 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

7. AAS – angle-angle-side
If two angles and a nonnonnonnon----includedincludedincludedincluded side of one
triangle are congruent to two angles and a
non-included corresponding 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
∆YOU ≅ ∆WIN

8. ASS – angle-side-side
(we do not cuss in math class)
There is no ASS (or SSA) congruence
theorem.
(unless the angle is a right angle — see next
slide)

9. 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
∆JOE ≅ ∆MAC