Similarity

Similarity Criteria

Using AA, SAS, and SSS to establish triangle similarity.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

When AAA Actually Is Enough

Recall from the congruence lesson that AAA alone couldn't prove two triangles congruent, since matching angles doesn't fix the size. Before reading on, think about this: if congruence needs more than just matching angles, could matching angles alone be exactly enough for a weaker relationship — same shape, but not necessarily the same size?

Definition — Similar Triangles

Two triangles are similar if a sequence of rigid motions and a dilation (a size change) maps one onto the other — equivalently, if their corresponding angles are congruent and their corresponding sides are proportional, sharing a constant scale factor.

Definition — AA, SAS, and SSS Similarity

AA similarity: two pairs of congruent corresponding angles guarantee similarity. SAS similarity: two proportional corresponding sides with a congruent included angle guarantee similarity. SSS similarity: three proportional corresponding sides guarantee similarity.

AA similarity works with only two angles because of a fact you already know: every triangle's three angles sum to exactly 180°. If two angles of one triangle match two angles of another, the third angle in each triangle is forced to be whatever's left over from 180° — meaning it's automatically the same in both triangles too. Matching just two angles secretly matches all three, and matching all three angles is exactly what fixes a triangle's shape (without fixing its size), which is the actual definition of similarity.

Worked Example — Proving Similarity with AA

Triangle ABC has angles 50° and 70°. Triangle DEF has angles 50° and 60°. Are they similar? Triangle ABC's third angle: 180 − 50 − 70 = 60°. Triangle DEF now has angles 50°, 60°, and 70° (matching all three angles from ABC, once ABC's third angle is found). Since all corresponding angles match, the triangles are similar by AA.

Worked Example — Setting Up a Proportion with SAS Similarity

Triangle ABC has sides AB = 6 and AC = 9, with a 40° angle between them. Triangle DEF has sides DE = 8 and DF = 12, with the same 40° angle between them. Check the ratios: 8/6 = 4/3, and 12/9 = 4/3. Since both ratios match and the included angle is congruent, the triangles are similar by SAS similarity, with a scale factor of 4/3.

Tip

When checking SAS or SSS similarity, always confirm the ratios are set up between genuinely corresponding sides — matching the wrong pair of sides across two triangles can accidentally suggest a false or misleading ratio.

Common Mistakes

Confusing similarity criteria with congruence criteria, expecting SAS similarity to require the included angle be congruent and both sides be equal rather than proportional.
Similarity criteria require sides to be proportional (sharing a common scale factor), not necessarily equal — only the included angle itself needs to be exactly congruent in SAS similarity.
Concluding similarity from only one matching angle, without checking a second one.
AA similarity genuinely needs two matching angle pairs — one alone doesn't determine the third angle in either triangle, so it isn't enough to guarantee the whole shape matches.

Key Takeaways

Similar triangles share the same shape (proportional sides, congruent angles) but not necessarily the same size.
AA similarity works because a triangle's third angle is always determined by the other two, since all three sum to 180°.
SAS and SSS similarity require proportional corresponding sides, sharing one consistent scale factor.

Summary

Similarity criteria establish when two triangles share the exact same shape. The next lesson puts that relationship to practical use, finding unknown lengths using proportional reasoning.

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