SideAngleSide

SideAngleSide, commonly abbreviated as SAS, is a criterion for triangle congruence in Euclidean geometry. It states that if two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another triangle, the two triangles are congruent.

The included angle is the angle formed by the two sides being compared. Formally, if AB = DE, AC = DF, and angle BAC = angle EDF, then triangle ABC is congruent to triangle DEF. The justification rests on the existence of a rigid motion (a rotation and translation) that maps one triangle onto the other while preserving both side lengths and angles.

Converse and limitations: SAS is a valid congruence criterion, meaning it guarantees congruence when the stated conditions hold. It is not true that two sides with a non-included angle (the SSA configuration) uniquely determine a congruent triangle; SSA can yield zero, one, or two possible triangles, so it does not guarantee congruence.

Relation to other criteria: SAS is one of the standard triangle congruence postulates, alongside SSS (three sides), ASA (two angles and the included side), AAS (two angles and a non-included side), and HL (hypotenuse-leg, for right triangles) in Euclidean geometry. These criteria are used to prove triangle congruence without requiring explicit construction.

Applications: SAS is widely used in geometric proofs, triangle congruence checks, geometric constructions, and surveying calculations where exact alignment and measurements are essential. It provides a reliable test for equality of triangles under rigid motions.