Congruence Criteria for Triangles

A triangle is a rigid shape once you fix enough of it. Three side lengths leave no wiggle room: hinge three fixed rods together and only one triangle can form, so SSS (side-side-side) locks it down. Three angles don't work the same way, because angles only control shape, not size — a triangle with angles 40, 60, 80 could be tiny or huge, so AAA only gives similarity, never congruence. Mixing sides and angles gets you back to a fixed size as long as you include enough sides: SAS (side-angle-side, the angle between two known sides), ASA (angle-side-angle, the side between two known angles), and AAS (angle-angle-side, the side outside the angles) all pin down one unique triangle.

Before naming a criterion, match corresponding parts in the same order the triangles are named. If triangle ABC is congruent to triangle DEF, A matches D, B matches E, C matches F, and the sides between corresponding vertices match too: AB with DE, BC with EF, AC with DF. Check whether the given angle sits between the two given sides (SAS) or outside them (AAS) — that placement is the whole test.

SSA fails because an angle not between the two known sides can swing the third side into two different positions, producing two different triangles with the same three measurements. This is why the hinge logic matters: congruence needs the angle's position locked relative to the sides, not just its size. These criteria set up everything else in the unit — once triangles are proven congruent or similar, the same corresponding-parts logic carries over to the trig ratios and the laws of sines and cosines used on non-right triangles.