Two triangles are congruent when they have the same shape AND the same size — corresponding sides equal, corresponding angles equal. There are exactly 5 standard methods for proving congruence, and choosing the right one depends on what you’ve been given. This guide walks through all 5 with worked examples and common pitfalls.
Each method names what’s required. The pattern in every name: each letter is either an “S” (a side is given equal) or an “A” (an angle is given equal).
Note what’s missing: there is no SSA postulate (the “donkey theorem” — it doesn’t always work because the same SSA can fit two different triangles). And there is no AAA postulate either — equal angles only prove triangles similar, not congruent.
If all three sides of one triangle equal all three sides of another, the triangles are congruent. Order matters when matching: the longest side in one must equal the longest in the other, etc.
Example. Triangle ABC has AB = 5, BC = 7, CA = 6. Triangle DEF has DE = 5, EF = 7, FD = 6. By SSS, △ABC ≅ △DEF.
When to use it: when you have all three side lengths and no angle info. Common in surveying, drafting, and rigid-frame engineering proofs.
If two sides and the angle between them are equal, the triangles are congruent. The angle MUST be the included one (between the two given sides), or the proof falls apart.
Example. △ABC: AB = 8, ∠B = 50°, BC = 10. △DEF: DE = 8, ∠E = 50°, EF = 10. By SAS (the 50° angle is between the 8 and 10 sides in both), △ABC ≅ △DEF.
Common mistake: using SSA — two sides and a NON-included angle. This is NOT a valid postulate (SSA can produce two different triangles, the “ambiguous case”). Always verify the angle is sandwiched between the two sides.
If two angles and the side between them are equal, the triangles are congruent. The third angle is automatically determined (angles in a triangle sum to 180°), and the remaining sides follow from the Law of Sines.
Example. △ABC: ∠A = 40°, AB = 6, ∠B = 80°. △DEF: ∠D = 40°, DE = 6, ∠E = 80°. By ASA, △ABC ≅ △DEF.
When ASA appears in proofs: often when parallel lines give you alternate-interior or corresponding angles “for free”, and you have one shared/given side. This is the most common postulate in textbook proofs that involve parallel lines or transversals.
Like ASA but the side is NOT between the two given angles. Still valid because once two angles are fixed, the third is too — and a single side then locks in the size.
Example. △ABC: ∠A = 30°, ∠B = 70°, BC = 9. △DEF: ∠D = 30°, ∠E = 70°, EF = 9. By AAS, △ABC ≅ △DEF.
ASA vs AAS: the only difference is whether the equal side sits between the two equal angles. ASA: included side. AAS: non-included side. Both prove congruence; some textbooks combine them as “AAS/ASA”.
For right triangles ONLY. If the hypotenuse and one leg of one right triangle equal the hypotenuse and one leg of another, the triangles are congruent.
Example. Right △ABC has ∠C = 90°, hypotenuse AB = 13, leg BC = 5. Right △DEF has ∠F = 90°, hypotenuse DE = 13, leg EF = 5. By HL, △ABC ≅ △DEF.
Why HL is special: it’s effectively SSA — but because we KNOW one angle is 90°, the ambiguous case can’t happen. The third side is determined by the Pythagorean theorem (12 in this example), so once hypotenuse + leg match, everything matches.
None of these are valid congruence postulates:
The Congruent Triangle Calculator will tell you which postulate applies given your inputs and walk through the proof step-by-step. For congruence proofs that involve parallel lines, the Congruent Triangles with Parallel Lines Calculator handles the alternate-interior-angle setup automatically.
Why is SSA not a congruence postulate? Because two different triangles can satisfy SSA — the angle leaves room for the unknown side to swing into two positions. The exception is HL (which is technically SSA with a right angle), where the right angle removes the ambiguity.
What does CPCTC mean? “Corresponding Parts of Congruent Triangles are Congruent.” Once you’ve proved two triangles congruent by one of the 5 methods, ANY pair of corresponding sides or angles can be declared equal. CPCTC is the standard final step of most proofs that conclude two segments or angles are equal.
Are similar triangles also congruent? Only if the scale factor is 1. Similar triangles share the same shape (proportional sides, equal angles), but congruent triangles share the same shape AND the same size. All congruent triangles are similar; not all similar triangles are congruent.