Similarity (geometry)

Similar figures are related by a scaling. Every pair of corresponding angles is congruent, and every pair of corresponding sides has the same ratio — the scale factor k. Similarity is written with the ~ symbol, as in △ABC ~ △DEF, and the order of the letters tells you which vertices correspond. Congruence is the special case where k = 1.

Three shortcuts prove triangles similar. AA requires only that two pairs of angles match, since the third pair must then match too. SSS similarity requires all three pairs of corresponding sides to share a common ratio. SAS similarity requires two pairs of proportional sides with the included angles congruent. AA is the workhorse on tests, and it is often available because parallel lines create equal corresponding angles, or because two triangles share an angle.

Scale factor affects length, area, and volume differently. If k is the ratio of corresponding sides, then the ratio of perimeters is also k, the ratio of areas is k², and the ratio of volumes of similar solids is k³. Doubling every dimension of a triangle multiplies its area by 4, not by 2 — a distinction tests probe constantly. Once similarity is established, unknown lengths follow from setting up a proportion between corresponding sides and cross-multiplying.

Similarity is a core topic on the AMC 8 and AMC 12, where it usually appears embedded in a larger figure and must be spotted before anything else can be computed, and on the Praxis Core Math exam, which tests it alongside angle relationships and congruence. Look for shared angles, parallel segments, and right triangles split by an altitude — all reliable sources of similar triangles.