Right Triangle Congruence Calculator

Right triangles get four specialized congruence theorems — HL, HA, LA, and LL — that simplify the general SSS / SAS / ASA / AAS framework. The reason is simple: every right triangle starts with one angle already known (the 90°). With one element pre-supplied, you need fewer measurements to lock down the whole triangle compared to a general triangle. This tutorial explains each of the four theorems, when to use each, and why HL is the only one that cannot be derived as a special case of the general triangle postulates.

Why right triangles get their own rules

For a general triangle, you need three independent pieces of information (three sides; or two sides + an included angle; or two angles + an included or non-included side) to prove congruence. The pattern SSA — two sides and an angle opposite one of them — is famously insufficient for general triangles because it admits the ambiguous case (zero, one, or two valid triangles).

For right triangles, the 90° angle is built in. The pattern simplifies: you only need to specify two more elements that lock down the rest. The four specialized theorems describe those minimal pairs.

The four right-triangle congruence theorems

HL — the unique postulate

Hypotenuse-Leg states: if the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, the two triangles are congruent.

Why this is special: HL is the right-triangle analog of SSA, which is not generally valid. SSA can produce zero, one, or two triangles. But when the angle in SSA is the right angle, the third side is forced (by the Pythagorean theorem) — so SSA in a right triangle collapses to SSS, which IS valid.

HL is the only one of the four right-triangle theorems that cannot be proved as a corollary of the general SSS/SAS/ASA/AAS rules without using the Pythagorean theorem. The others are direct restatements with the 90° angle implicit.

LL — Leg-Leg (= SAS)

If both legs of one right triangle are equal to both legs of another, the triangles are congruent. The two legs are the two sides meeting at the right angle, so the right angle is the included angle — this is just SAS with the included angle pre-specified as 90°.

LA — Leg-Angle

If one leg and one acute angle (and the right angle, implicit) of one right triangle equal those of another, the triangles are congruent. This is ASA if the leg is between the two angles, or AAS if the acute angle is opposite the given leg. In either flavor, three pieces of information (leg, right angle, acute angle) lock down the triangle.

HA — Hypotenuse-Angle

If the hypotenuse and one acute angle (plus the implicit right angle) match between two right triangles, they are congruent. This is AAS — the hypotenuse is the side opposite the right angle, and the other angle is the acute angle, so we have AAS with the hypotenuse playing the "side opposite a given angle" role.

Worked example — using HL

Triangle 1: legs 6 and 8, hypotenuse 10 (a 6-8-10 right triangle = 2 × the 3-4-5).
Triangle 2: legs 6 and ?, hypotenuse 10.

By HL, the two triangles must be congruent — we have hypotenuse (10 = 10) and one leg (6 = 6). The unknown leg of Triangle 2 is therefore 8 by congruence (or, equivalently, by the Pythagorean theorem: √(100 − 36) = 8).

Worked example — using LL

Triangle 1: legs 5 and 12.
Triangle 2: legs 5 and 12.

By LL (= SAS with the included 90°), the triangles are congruent. Hypotenuse: √(25 + 144) = √169 = 13 for both. This is the 5-12-13 right triangle.

Why this matters for proofs

In two-column proofs, every step needs a justification. "HL", "LL", "LA", and "HA" are accepted justifications that save you from writing out the full SSS/SAS/ASA/AAS chain. Many textbook problems explicitly require you to use one of these abbreviated forms.

The AI Solve button on this calculator generates a step-by-step proof using whichever of the four theorems is appropriate for your inputs — useful for checking a proof you wrote by hand or for understanding why two triangles you suspect are congruent really are.

Common mistakes

• Using "SSA" naming for HL. Right-triangle HL is essentially SSA with the right angle, but the right-triangle context is what makes it valid. Just calling it "SSA" without the right-angle qualifier is wrong.
• Treating LA without specifying the angle. "Leg-Angle" is ambiguous if there are two acute angles — be specific about which acute angle is being matched. Both acute angles must match for the triangles to be congruent (since they sum to 90°, knowing one determines the other).
• Forgetting the right angle is needed. All four theorems require both triangles to be right triangles. Two non-right triangles with matching legs and hypotenuse are not necessarily congruent.
• Confusing congruence with similarity. Congruent triangles are identical (same shape AND same size). Similar triangles have the same shape but may differ in scale. HL/HA/LA/LL prove congruence; for similarity, use AA / SSS-similarity / SAS-similarity.