Triangle Congruence & Similarity: SSS, SAS, ASA, AA and the Famous Traps
When are two triangles identical, when are they just scaled copies, and why do AAA and SSA never prove congruence? The five valid rules, the AA shortcut for similarity, and the k² / k³ area-and-volume scaling rule.
Same shape vs. same size
Two triangles in front of you. Are they identical? Are they just scaled copies? Or are they unrelated? The Texas Geometry CBE asks this kind of question on roughly one in eight problems — and the answer comes down to two precise vocabulary words.
Congruent = same shape and same size (a perfect copy). Similar = same shape, possibly different size (a proportional copy). Every triangle question hinges on knowing which one you're being asked about.
- Congruent (≅)
- All corresponding sides equal and all corresponding angles equal. Triangles match perfectly.
- Similar (~)
- All corresponding angles equal and sides proportional. Same shape, scaled.
- Scale factor
- The ratio of corresponding sides between similar figures. If 5 → 15, the scale factor is 3.
Congruence: the five rules
You don't have to verify all six parts (3 sides + 3 angles) to prove triangles are congruent. There are exactly five shortcuts:
AAA (three angles match) tells you the triangles are similar — not congruent. They could be different sizes. SSA (two sides and a non-included angle) is the famous “ambiguous case” — it can describe two completely different triangles. Both appear in “which CANNOT prove congruence?” questions.
Pick the right shortcut
Two triangles have all three pairs of sides equal. By which congruence rule are they congruent?
Open the question →Spot the trap
Which of the following is NOT a valid way to prove two triangles congruent?
Open the question →Similarity: the proportion machine
Two figures are similar when one is a scaled copy of the other. The angles match exactly; the sides are proportional. To find a missing side, set up a proportion of corresponding sides.
Worked example
A rectangle measures 12 by 8. A similar rectangle has width 6. What is its length?
Set up a proportion
A rectangle is 12×8. A similar rectangle has width 6. What is its length?
Open the question →The fastest similarity rule: AA
For triangles only, you don't need three angles — just two. If two angles of one triangle equal two angles of another, the triangles are similar (the third angle is forced because all triangle angles sum to 180°).
Two pairs of equal angles ⇒ similar triangles. This is the most-used similarity tool on the test — especially in “shadow” word problems where the sun's angle creates two similar right triangles.
Shadow problems — AA in disguise
Classic CBE setup: a tree casts a shadow, a person casts a shadow at the same time, find the height. The two right triangles share the sun's angle of elevation, plus both have a 90° angle — that's two pairs of equal angles, so the triangles are similar.
The shadow problem
A tree casts a shadow 18 feet long. A 5-foot fence post casts a shadow 6 feet long. How tall is the tree?
Open the question →The square-of-the-scale-factor rule
If two similar figures have a side-length scale factor of k, then their areas are in the ratio k² and their volumes are in the ratio k³. This is the highest-yield rule on similarity word problems.
3-second recap
- Congruent ≅ — same shape & size. Use SSS, SAS, ASA, AAS, HL only.
- Similar ~ — same shape, scaled. Set up a proportion of corresponding sides.
- AA proves similarity for triangles (because the third angle is forced).
- AAA & SSA never prove congruence — classic CBE traps.
- Side ratio k → area ratio k² → volume ratio k³.