Proving Congruence: ASA, AAS & HL
The angle-based criteria, and the one reserved for right triangles.
The explanation
Three more ways to prove triangles congruent.
ASA: two angles and the side *between* them.
AAS: two angles and a side *not* between them. This works because knowing two angles gives you the third for free, which turns it back into ASA.
HL: for right triangles only — the hypotenuse and one leg.
HL is the exception that makes SSA work, and only because the right angle removes the ambiguity.
So the full list is SSS, SAS, ASA, AAS and HL. The two that do not work are SSA (except as HL) and AAA.
To use HL you must state that both triangles are right triangles. Skipping that step invalidates the proof.
ASA is typically a postulate; AAS follows from it as a theorem via the Triangle Sum Theorem, since two pairs of congruent angles force the third pair to be congruent, converting the non-included side into an included one.
HL applies only to right triangles and is provable from the Pythagorean theorem: with the hypotenuse and one leg fixed, the remaining leg is determined as √(c² − a²), reducing HL to SSS. This is why the right angle rescues what would otherwise be the ambiguous SSA configuration.
The complete set of valid criteria is SSS, SAS, ASA, AAS and HL. AAA establishes similarity only. SSA is invalid in general.
The choice among them is driven by the diagram: two marked angles with a side between them give ASA, a side outside them gives AAS, and a right-angle mark with hypotenuse and leg gives HL. When a diagram supplies parallel lines, the alternate interior angle theorem is usually the source of the second angle pair.
Worked example
Given: ∠A ≅ ∠D, ∠B ≅ ∠E, and BC ≅ EF. Which criterion proves △ABC ≅ △DEF?
- BC is opposite ∠A, and EF is opposite ∠D.
- So the congruent side is not between the two congruent angles.
- Two angles and a non-included side.
Answer: AAS (equivalently ASA, since the third angle pair follows from the Triangle Sum Theorem).
Common mistakes
- Applying HL without first establishing that both triangles have a right angle.
- Labelling an AAS configuration as ASA, or vice versa, by misreading which side is included.