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. For education statistics and research, visit the National Center for Education Statistics.

To use this theorem: For education statistics and research, visit the National Center for Education Statistics.

• 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? For education statistics and research, visit the National Center for Education Statistics.

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. For education statistics and research, visit the National Center for Education Statistics.

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? For education statistics and research, visit the National Center for Education Statistics.

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. For education statistics and research, visit the National Center for Education Statistics.

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: For education statistics and research, visit the National Center for Education Statistics.

1. Yes
2. No