AngleAngleSide

Angle-Angle-Side, often abbreviated as AAS, is a triangle congruence criterion used in geometry. It states that if two angles of one triangle are equal to the corresponding two angles of another triangle and the side opposite one of those equal angles is also equal to its corresponding side, then the two triangles are congruent. The side referenced is not the side between the two given angles.

AAS is a valid and commonly used criterion for establishing triangle congruence. It is effectively equivalent to the ASA (Angle-Side-Angle) criterion because two angles determine the third angle, and the known side together with the two equal angles fixes the triangle’s shape and size. Many proofs use AAS directly or convert it to ASA when convenient.

In practice, AAS allows one to deduce all corresponding parts of the triangles: remaining angles are equal and the other two sides are proportional to the given congruent side, establishing full congruence. The criterion is not the same as SSA (two sides and a non-included angle), which can be ambiguous in some cases and does not constitute a general congruence rule.

Examples in geometry problems often involve showing ∠A = ∠A', ∠B = ∠B', and a common or corresponding side a = a', leading to the conclusion that triangles ABC and A'B'C' are congruent. This permits immediate inference of equal remaining angles and sides.

See also: ASA, SSA, triangle congruence.