Understanding Triangle Congruence
Two triangles are congruent if they have exactly the same size and shape. This means all corresponding sides are equal in length and all corresponding angles are equal in measure. However, you do not need to know all six measurements to prove congruence.
Congruence Criteria
SSS (Side-Side-Side)
All three pairs of corresponding sides are equal.
a1=a2, b1=b2, c1=c2
SAS (Side-Angle-Side)
Two pairs of sides and their included angle are equal.
a1=a2, C1=C2, b1=b2
ASA (Angle-Side-Angle)
Two pairs of angles and their included side are equal.
A1=A2, c1=c2, B1=B2
AAS (Angle-Angle-Side)
Two pairs of angles and a non-included side are equal.
A1=A2, B1=B2, a1=a2
HL (Hypotenuse-Leg)
For right triangles: the hypotenuse and one leg are equal.
hyp1=hyp2, leg1=leg2
Important Notes
- SSA (Side-Side-Angle) is NOT a valid congruence criterion due to the ambiguous case.
- AAA (Angle-Angle-Angle) proves similarity but not congruence.
- HL only applies to right triangles.
- Congruent triangles have equal areas and perimeters.