SAS and SSS similarity
Alongside AA, there are two more ways to confirm that two triangles are similar, and these ones bring side lengths into the test. They are the similarity cousins of the SAS and SSS congruence criteria: same letters, but now the sides need only be in proportion rather than equal.
SSS similarity: if all three pairs of corresponding sides are in the same ratio — AB/DE = BC/EF = CA/FD — then the triangles are similar. You check three side-ratios and confirm they all reduce to one common scale factor. SAS similarity: if two pairs of corresponding sides are in proportion AND the angles squeezed between those two sides are equal, then the triangles are similar. The angle must be the included angle (the one between the two measured sides); an equal angle somewhere else does not qualify, just as it does not for SAS congruence.
When do you reach for these instead of AA? Use them when a problem hands you lengths rather than angles. If you are told the three sides of each triangle, SSS settles similarity immediately; if you have two sides and the angle between them, SAS does. They are also the criteria that let you scale a triangle by a known factor and be certain the result is similar. As always, the catch is correspondence: match the longest side to the longest, the shortest to the shortest, and keep the included angle truly between the matched sides.
Triangle with sides 3, 4, 5 and triangle with sides 9, 12, 15: the ratios are 3/9 = 4/12 = 5/15 = 1/3, all equal, so by SSS they are similar with scale factor 3. No angle measurement needed.
Three equal side-ratios (SSS) clinch similarity without touching a protractor.
The angle in SAS similarity must be the included angle, between the two proportional sides. The non-included version, 'SSA', fails for similarity for the same reason it fails for congruence — it can describe two genuinely different triangles.