Congruent Triangles

Two triangles are congruent if all six corresponding parts (three sides and three angles) of the triangles are congruent. In layman’s terms, two triangles are congruent if they have same size and same shape.

Proving Two Triangles Are Congruent

However, to prove two triangles are congruent, we only need to show three parts (combination of sides and/or angles) of one triangle is congruent to the corresponding three parts of another triangle. The three parts can be side-angle-side (SAS), angle-side-angle (ASA), side-side-side (SSS) or angle-angle-side (AAS).

	SAS If two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the two triangles are congruent.
	ASA If two angles and the included side of one triangle are congruent to the corresponding two angles and the included side of another triangle, then the two triangles are congruent.
	SSS If all three sides of one triangle are congruent to the corresponding three sides of another triangle, then the two triangles are congruent.
	AAS If two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, then the two triangles are congruent.

However, you can’t prove two triangles are congruent by showing all three angles of one triangle are congruent to the corresponding three angles of another triangle. For example, the corresponding angles of two triangles shown below have same degree measures, but obviously  the triangle on the left is larger than the one on the right. On the other hand, they still have the same shape. In this case, we say the two triangles are similar.

You also can’t prove two triangles are congruent by showing two sides and a non-included angle of one triangle are congruent to the corresponding two sides and non-included angle of another triangle (see below).

In conclusion, we can prove two triangles are congruent by SAS, ASA, SSS and AAS, and we can’t prove two triangles are congruent by AAA and SSA.