The big idea SAS and SSS appear in both congruence and similarity. Equal sides prove congruence; proportional sides prove similarity.
SAS, twice
SAS congruence: two sides and their included angle equal the matching two sides and included angle. The scale factor is 1, so the figures are congruent and therefore also similar.
SAS similarity: one pair of included angles is congruent, and the two pairs of sides around those angles are proportional. The scale factor can differ from 1.
The labeled sides must be the two sides that form the marked angle. A true proportion on the wrong side pair does not prove SAS similarity.
Lookalike: if sides 5 and 3 match sides 5 and 3, congruence is the stronger name. If sides 2 and 8 match sides 1 and 4, the triangles are similar but not congruent.
SSS, twice
SSS congruence needs three pairs of equal sides. SSS similarity needs three pairs of proportional sides with one scale factor for all three.
Keep one triangle in every numerator. Mixing 8/2 with 3/12 flips one pair and ruins the comparison.
With no angle marks, check SSS similarity. With two angle marks, AA is faster.
Nested triangles
Suppose a small triangle shares angle A with a larger triangle and ΔABC ~ ΔADE. Then AD/AB = AE/AC.
If AD = 8, AB = 14, and AE = 4, then 8/14 = 4/AC, so AC = 7.
Do not pair an outer leftover segment with a whole side. The similarity statement tells you which lengths correspond.
Your turn
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