Three ways to prove two triangles are the same shape, even if they're different sizes.
The Mnemonic
Same Shape, Different Size
Two triangles are similar (symbol ~) if they have the exact same shape — all three corresponding angles are equal — even if their sizes are completely different. Sides don't have to be equal for similar triangles; they only have to be proportional, meaning every side of one triangle is the same multiple of the corresponding side in the other.
There are three shortcuts to prove similarity, echoing the congruence shortcuts but with looser requirements: AA (two angles match — the third is automatically forced to match too, by the Triangle Angle Sum Theorem), SAS~ (two sides proportional with the same included angle), and SSS~ (all three sides proportional).
△ABC ~ △DEF — same shape, different size. Angles match exactly; sides are proportional, not equal
💡 Memory Trick
"Three ways to prove triangles similar." AA needs the least information (just 2 angles) because a triangle's third angle is never free to vary once two are fixed. SAS~ and SSS~ both require checking that sides are proportional — meaning you divide corresponding sides and confirm you get the same ratio (scale factor) every time, not that the sides are literally equal.
Why It Works
Similarity vs. Congruence — Not the Same Thing
It's tempting to treat similarity as "a weaker version" of congruence, and that's basically correct — congruent triangles are always similar (they're identical, so obviously proportional with a scale factor of exactly 1), but similar triangles are only sometimes congruent (only when that scale factor happens to be 1).
This connects directly back to Congruence Shortcuts: recall that AAA was explicitly ruled out as a congruence shortcut, because matching all three angles only guarantees the same SHAPE, not the same size. That's not a coincidence — AAA (or really, just AA, since the third angle is automatic) is precisely the similarity shortcut that congruence needed to reject.
Using It In A Proof
Using Similarity to Solve for Missing Lengths
Once similarity is established, the real power is setting up a proportion between corresponding sides to solve for an unknown length — this is by far the most common use of similarity in practice.
1
Establish similarity first
Prove the triangles similar using AA, SAS~, or SSS~ before doing anything else — you can't set up a valid proportion between triangles that haven't been shown to be similar.
2
Match up corresponding sides carefully
Corresponding sides are determined by the similarity statement's letter order, exactly like with congruence — △ABC ~ △DEF means AB corresponds to DE, BC to EF, and AC to DF.
3
Set up and solve the proportion
Write corresponding sides as equal ratios (a cross-multiplication problem), then solve for the unknown.
Full Worked Example
Solving for a Missing Side Using AA Similarity
Given: △ABC ~ △DEF (established by AA). AB = 6, BC = 8, DE = 9. Find: EF.
1
Confirm the correspondence
Since △ABC ~ △DEF, AB corresponds to DE, and BC corresponds to EF.
2
Set up the proportion
AB/DE = BC/EF, so 6/9 = 8/EF.
3
Cross-multiply
6 × EF = 9 × 8, so 6 × EF = 72.
4
Solve
EF = 12.
Sanity check: the scale factor from △ABC to △DEF is 9/6 = 1.5. Applying that same factor: 8 × 1.5 = 12. ✓ Matches.
🎯 Quick Worked Example
Two triangles share ∠A, and AB/AD = AC/AE = 2/3. Which similarity shortcut applies?
1
Identify what's given. One shared angle (∠A), and two pairs of sides with the SAME ratio (2/3) — and ∠A is the included angle between those two sides in both triangles.
2
Match to a shortcut. Two proportional sides with the included angle between them matches SAS~ exactly.
3
Conclude. The triangles are similar by SAS~ Similarity.
📌 Exam Application
AA is the fastest similarity shortcut to spot on an exam — two triangles sharing a common angle (often from a shared vertex or parallel lines creating equal corresponding/alternate angles) plus one more matching angle is all that's needed, and it comes up far more often than SAS~ or SSS~ in typical problems.
⚠️ Most Common Triangle Similarity Mistakes
Trap 1 — Assuming proportional means equal: Similar triangles need proportional sides, not equal sides — dividing corresponding sides should give the same ratio (scale factor) throughout, not the same length.
Trap 2 — Matching up the wrong corresponding sides: Just like with congruence, the similarity statement's letter order determines correspondence — always re-derive which sides match from the statement itself (△ABC ~ △DEF) rather than assuming based on how the diagram looks.
✓ Quick Self-Test
1) What is the key difference between similar and congruent triangles? 2) Why does AA only need two angles instead of three? 3) If △PQR ~ △STU with PQ = 4, ST = 10, and QR = 6, find TU. 4) Are all congruent triangles similar? Are all similar triangles congruent? 5) Which similarity shortcut requires all three sides to be proportional?