Right Triangle Similarity and Congruence: the HL Theorem

An interactive demonstration of the Hypotenuse-Leg (HL) theorem for right triangles. Shows both HL congruence and HL similarity, which are special cases that work for right triangles but not for general triangles.

Prerequisites: triangle-congruence, triangle-similarity · Difficulty: intermediate

The Hypotenuse-Leg (HL) theorem is a special case that works only for right triangles: if the hypotenuse and one leg of one right triangle are equal (or proportional) to those of another, the triangles are congruent (or similar).

Strategy: show two right triangles sharing a right angle, an equal (or proportional) hypotenuse, and one equal (or proportional) leg; then use the Pythagorean theorem to recover the third side, giving SSS congruence or similarity.

HL Congruence

The Hypotenuse-Leg (HL) theorem is a congruence test for right triangles: if the hypotenuse and one leg of a right triangle equal those of another, the triangles are congruent.

Why HL Congruence Works

By the Pythagorean theorem, if the hypotenuse and one leg are equal, the remaining leg must also be equal — giving SSS congruence.

HL Similarity

The HL similarity theorem extends the idea: if the hypotenuse and one leg of a right triangle are proportional to those of another, the triangles are similar.

Why HL Similarity Works

By the Pythagorean theorem, if the hypotenuse and one leg scale by $k$, the remaining leg must scale by $k$ too — giving SSS similarity.

Summary — HL Theorem

The HL theorem is SSA for right triangles. The right angle eliminates the ambiguity that makes SSA fail for general triangles.

• HL Congruence: Equal hypotenuse + Equal leg → Congruent
• HL Similarity: Proportional hypotenuse + Proportional leg → Similar

The Hypotenuse-Leg (HL) theorem is a powerful tool specifically for right triangles:

• HL Congruence: Right angle + Equal hypotenuse + Equal leg → Triangles are congruent
• HL Similarity: Right angle + Proportional hypotenuse + Proportional leg → Triangles are similar

Key Insight

HL is essentially SSA, but the right angle removes the ambiguity that makes SSA fail for general triangles. By the Pythagorean theorem, if the hypotenuse and one leg are equal (or proportional), the other leg must also be equal (or proportional) — so all three sides match!

Notes

For general triangles, SSA (Side-Side-Angle) is ambiguous: the same three measurements can produce two different triangles. HL is the special case of SSA that DOES work, because the right angle ($90°$) removes that ambiguity. Once the hypotenuse, one leg, and the right angle are fixed, exactly one triangle fits.

HL Congruence

For two right triangles to be congruent by HL:

1. Both triangles must be right triangles
2. The hypotenuses are equal in length
3. One pair of corresponding legs are equal in length

HL Similarity

For two right triangles to be similar by HL:

1. Both triangles must be right triangles
2. The hypotenuses are proportional (have the same ratio)
3. One pair of corresponding legs are proportional (same ratio)