Congruent Triangles - Two angles and an opposite side (AAS)

Definition: Triangles are congruent if two pairs of corresponding angles and a pair of opposite sides are equal in both triangles.

There are five ways to test that two triangles are congruent. This is one of them (AAS). For a list see Congruent Triangles. If there are two pairs of corresponding angles and a pair of corresponding opposite sides that are equal in measure, then the triangles are congruent.

By opposite side we mean a side opposite either one of the angles. Either one will do, but it has to be the same one in both triangles obviously. In the figure below we have chosen the side QR which is opposite the angle P.

Try this Drag any orange dot at P,Q,R. The other triangle LMN will change to remain congruent to the triangle PQR.
Notice that the the two angles and the opposite side are drawn in thick blue lines to indicate they are the parts being used to test for congruence.

What does this mean?

Because the triangles are congruent:
  1. the other two sides are equal (PQ=LM, and PR=LN)
  2. the third angles a re equal (Q=M)

But don't forget:

Two triangles, congruent because all three sides are the same length, but one flipped and rotated Congruent triangles can be rotated and/or mirror images of each other (reflected). (See Congruent triangles.) In the figure above, the two triangles have all three corresponding sides equal in length and so are still congruent, even though one is the mirror image of the other and rotated.

Other congruence topics

Congruent Triangles

Congruent Polygons