Congruence of Right Triangles

Congruence Criteria for Right Triangles

Right triangles have four congruence criteria (fewer than general triangles, since one right angle is already known):

Criterion 1: Two Legs (LL) — If the legs of one right triangle are respectively equal to the legs of another, then the triangles are congruent (by SAS with the included right angle).

Criterion 2: Hypotenuse and One Leg (HL) — If the hypotenuse and one leg of one right triangle are respectively equal to the hypotenuse and one leg of another, then the triangles are congruent. This criterion is unique to right triangles.

Criterion 3: One Leg and an Adjacent Acute Angle (LA) — If a leg and the acute angle adjacent to that leg in one right triangle are respectively equal to those in another, then the triangles are congruent (by ASA).

Criterion 4: Hypotenuse and an Acute Angle (HA) — If the hypotenuse and one acute angle are respectively equal, then the triangles are congruent (by AAS, since the right angle provides the second angle).

Important Notes

For general triangles, knowing two sides and a non-included angle (SSA) does not guarantee congruence, but for right triangles the right angle removes the ambiguity. A common mistake is using SSA for general triangles — only right triangles permit the HL shortcut.