Congruent Triangles

What "congruent" means

Two shapes are congruent when they are identical — exactly the same shape and exactly the same size. If you could slide, turn or flip one of them, it would land perfectly on top of the other, covering it with no gaps and no overlap.

We write congruence with the symbol \cong. So two triangles being congruent looks like this:

\triangle ABC \cong \triangle DEF

When two triangles are congruent, every corresponding part matches: each pair of corresponding sides is equal in length, and each pair of corresponding angles is equal.

The four tests

You don't need to check all six pairs of parts. Just three matching parts, chosen the right way, are enough to force two triangles to be congruent. There are exactly four such tests.

Two triangles are congruent if any one of these matches:

SSS — all three pairs of sides are equal.
SAS — two pairs of sides and the included angle (the angle between those two sides) are equal.
ASA — two pairs of angles and the included side (the side between those two angles) are equal.
RHS — in two right-angled triangles, the right angle, the hypotenuse and one other side are equal.

Notice what is missing: AAA is not a congruence test. Three equal angles guarantee the same shape, but the triangles can be different sizes (think of a small triangle and a scaled-up copy) — that is similarity, not congruence.

See the matching parts

Here are two identical (congruent) triangles. Step through to see which three parts each test relies on — equal sides get matching tick marks, equal angles get matching arcs, and the right angle gets a square.