Triangle Similarity: AA, SSS & SAS

Triangles are easier to prove similar than other polygons, because you do not need both conditions.

AA: two pairs of congruent angles. That is enough. The third pair must match because all three sum to 180°.

SSS similarity: all three pairs of sides in the same ratio.

SAS similarity: two pairs of sides in the same ratio with the included angles congruent.

AA is the one you will use most, and parallel lines are its usual source — they hand you congruent corresponding or alternate interior angles for free.

This is the machinery behind indirect measurement. A person and a flagpole cast shadows at the same time, forming two similar triangles, and the flagpole's height comes from a proportion.

AA similarity holds because the Triangle Sum Theorem makes the third angle pair automatically congruent, and equiangular triangles have proportional sides. It is the criterion that fails as a congruence criterion (AAA) precisely because it fixes shape without size.

SSS similarity requires a₁/a₂ = b₁/b₂ = c₁/c₂, and SAS similarity requires two proportional pairs with congruent included angles — the same included-angle caution as in SAS congruence.

Setting up the proportion correctly matters more than the algebra. Corresponding sides must occupy corresponding positions, which the similarity statement's vertex order determines. Writing the proportion as (side of small)/(matching side of large) consistently on both sides prevents inversion errors.

Similar triangles are the engine behind several later results: the Side-Splitter Theorem, the geometric mean relations in right triangles, the constancy of trigonometric ratios, and the invariance of slope. Each is an application of the same criterion.