Proving Triangles Congruent

Overview

Proving two triangles congruent does not require showing that all six pairs of parts match. Five shortcuts — SSS, SAS, ASA, AAS, and HL — each name a minimum set of congruent parts that is enough to guarantee the triangles are congruent. The name of each shortcut is critical: it tells you exactly which parts you need and where they must sit in the picture, such as the angle included between two sides for SAS. The working strategy is process of elimination — list the parts you can show congruent, then check which shortcut they satisfy.

Vocabulary

SSS (Side-Side-Side): If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
Included Angle: The angle formed by two given sides — the angle that lies between them, sharing their common vertex.
ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. The included side is the side between the two angles.
AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
HL (Hypotenuse-Leg): If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. HL works only for right triangles.

Main Ideas

Only certain minimum sets of parts prove congruence: SSS, SAS, ASA, AAS, and HL (for right triangles only).
Position matters as much as the parts themselves: SAS needs the angle included between the two sides, ASA needs the side included between the two angles, and AAS needs the side in the corresponding non-included position.
SSA (two sides and a non-included angle) and AAA are not valid shortcuts — they do not determine one unique triangle.
Work by elimination: mark the congruent parts in the picture, name the corresponding parts, then test the marked parts against the list of shortcuts.
Free information often completes a shortcut: a shared side is congruent to itself (reflexive), vertical angles are congruent, and a midpoint creates two congruent segments.
Once congruence is proved, write the congruence statement with the vertices in corresponding order and name the shortcut used.