More triangle congruence criteria: AAS, HL, and why SSA fails

Besides SSS, SAS, and ASA, two more conditions prove triangles congruent. Angle-Angle-Side (AAS) works because two pairs of congruent angles force the third angles to match too, since each triangle's angles sum to 180 degrees, which turns AAS into ASA. Hypotenuse-Leg (HL) works only for right triangles: if the hypotenuses and one pair of legs are congruent, the Pythagorean theorem forces the other legs to be equal, so the triangles are congruent by SSS. Side-Side-Angle (SSA), where the angle is not between the two sides, does not always work, because the same two sides and angle can sometimes form two different triangles, and Angle-Angle-Angle (AAA) proves only that triangles are similar, not the same size. Once triangles are proved congruent, all their other corresponding parts are congruent too, often abbreviated CPCTC (corresponding parts of congruent triangles are congruent).

This explanation is AI-generated and can occasionally be wrong. 🚩 Report