18.4 Congruence and congruence criteria — Learn

Prerequisites: triangle angle facts, side lengths and equality.

Objectives: define congruence and choose a valid triangle congruence criterion.

Two shapes are congruent when they have the same shape and the same size. You can place either shape exactly over the other after a reflection, rotation or translation.

For triangles, use the given matching sides and angles to establish congruence. Common criteria are SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), and RHS (right angle, hypotenuse and one side).

Example. Two triangles have side lengths $5$, $7$ and $9$ units each.

All three corresponding sides match. Therefore SSS proves the triangles congruent.

Example. Two right-angled triangles have equal hypotenuse and one matching leg.

Both triangles have a right angle. The hypotenuses match, and one other side matches. Therefore RHS proves congruence.

Watch out. Three equal angles show the same shape, but not the same size. For example, triangles with sides $3,4,5$ and $6,8,10$ have equal angles but are not congruent.

Progress check

Could two mural triangles with matching angles but different side lengths cover one another exactly? Explain why not.

You should now be able to

• explain what congruent shapes are;
• identify SSS, SAS, ASA and RHS information;
• explain why three matching angles do not prove congruence;
• use a valid criterion to justify that triangles are congruent.