Geometry A hub · 03.05

Congruence and similarity together

Same letters. Equal vs proportional. Congruence is similarity with k = 1.

SAS, twice

SAS congruence: two sides and the included angle equal the matching two sides and included angle. Scale factor 1. Figures congruent, hence also similar.

SAS similarity: one pair of congruent included angles, and the two sides that make those angles proportional. Scale factor can be anything. Figures similar, congruent only if that factor is 1.

The sides have to be the ones that create the marked angle. If the labeled lengths are not those two sides, you cannot name SAS similarity even if an angle matches.

Lookalike pair: a quiz will offer SAS congruence and SAS similarity for the same picture. If 5 and 3 match 5 and 3, both are true and congruence is the stronger name. If 2 and 8 match 1 and 4, only similarity.

SSS, twice

SSS congruence: three pairs of equal sides. SSS similarity: three pairs of proportional sides, one k for all three.

Keep one triangle in the numerators. Mixing 8/2 with 3/12 is a flipped pair and will fail the check even when the triangles are similar.

If you have no angle marks, SSS similarity is the route. If you have two angle marks, AA is faster.

Nested triangles

A small triangle sharing angle A with a big one, ΔABC ~ ΔADE. Then AD/AB = AE/AC. If AD = 8, AB = 14, AE = 4, then 8/14 = 4/AC, so AC = 7.

Do not pair the leftover outer segment with a whole side. The similarity statement tells you which lengths belong together.

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