# HL Congruence: The Special Case of Right Triangles

Right triangles are distinct due to their one right angle. This uniqueness also translates to their congruence criteria. While other triangles rely on combinations of sides and angles, right triangles have a special shortcut: the HL Congruence theorem. This theorem deals with the hypotenuse and one leg of a right triangle, providing a swift way to determine congruence. Let's unfold this unique postulate!

## Step-by-step Guide: HL Congruence

**HL (Hypotenuse-Leg) Congruence Theorem**:

If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.

To use this theorem:

- Ensure that both triangles in question are right triangles.
- Identify and compare the lengths of the hypotenuses and one of the legs of both triangles.
- If both the hypotenuse and one leg are of equal length in the two triangles, then they are congruent by the HL theorem.

### Examples

**Example 1:**

Are right triangles with a hypotenuse of \(13 \text{ cm}\) and one leg of \(5 \text{ cm}\) and another right triangle with a hypotenuse of \(13 \text{ cm}\) and one leg of \(5 \text{ cm}\) congruent based on the HL Congruence theorem?

**Solution**:

Both triangles have:

A hypotenuse of \(13 \text{ cm}\)

One leg of \(5 \text{ cm}\)

Since the hypotenuse and one leg match in both triangles, they are congruent by the HL Congruence theorem.

**Example 2:**

Are right triangles with a hypotenuse of \(15 \text{ cm}\) and one leg of \(9 \text{ cm}\), and another right triangle with a hypotenuse of \(15 \text{ cm}\) and one leg of \(12 \text{ cm}\) congruent based on the HL Congruence theorem?

**Solution**:

The triangles have:

A matching hypotenuse of \(15 \text{ cm}\)

But differing leg lengths of \(9 \text{ cm}\) and \(12 \text{ cm}\)

Since one of the legs doesn’t match, the triangles are not congruent by the HL Congruence theorem.

### Practice Questions:

- Are right triangles with a hypotenuse of \(17 \text{ cm}\) and one leg of \(8 \text{ cm}\), and another triangle with a hypotenuse of \(17 \text{ cm}\) and one leg of \(8 \text{ cm}\) congruent by the HL Congruence theorem?
- Are right triangles with a hypotenuse of \(20 \text{ cm}\) and one leg of \(16 \text{ cm}\), and another triangle with a hypotenuse of \(20 \text{ cm}\) and one leg of \(15 \text{ cm}\) congruent by the HL Congruence theorem?

**Answers**:

- Yes
- No

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