HL Congruence: The Special Case of Right Triangles

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:

1. 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?
2. 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:

1. Yes
2. No