Congruent Triangles

SSS, SAS, ASA, AAS, HL — and why SSA isn't enough.

The idea

Two triangles are congruent when one can be moved — slid, turned, or flipped — to lie exactly on the other: the same shape and the same size. Congruence means all six pairs of corresponding parts agree, three pairs of sides and three pairs of angles.

A congruence criterion is a short list of matching parts that already forces all six pairs to agree, because only one triangle can be built from the parts it names. Three sides is the model case: three given lengths close into a triangle in exactly one way, so when the sides agree the angles must agree as well.

Whether a list qualifies depends on how its parts are arranged, not only on how many there are — an angle between two given sides determines the triangle, while the same angle placed elsewhere may not. A criterion proves congruence from a few matching parts; once congruence is established, every remaining pair of corresponding parts is equal, since congruence is an exact overlay.

The figure shows congruent triangles $ABC$ and $DEF$ in different orientations: matching tick marks pair the equal sides, and matching arcs pair the equal angles.

Ways to work on it

• Walkthrough. Define congruence, name each criterion, and conclude equal corresponding parts.
• Practice. Pick the congruence criterion that matches the given parts.
• Hardest. Why side-side-angle fails, and what extra fact rescues a proof.

Not sure where to start? Take the ten-question placement test.