# Congruent Triangles - Hypotenuse Leg Theorem

In these lessons, we will learn

- the Hypotenuse-Leg Theorem
- why the Hypotenuse-Leg Theorem is enough to prove triangles congruent
- the proof of the Hypotenuse-Leg Theorem using a two-column proof
- how to prove triangle congruence using the Hypotenuse-Leg Theorem

Related Topics: More Geometry Lessons

## Hypotenuse Leg Theorem

*Hypotenuse Leg Theorem *is used to prove whether a given set of right triangles are congruent.

**The Hypotenuse Leg (HL) Theorem states that**

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

In the following right triangles Δ*ABC* and Δ*PQR* , if *AB* = *PR*, *AC = QR *then Δ*ABC* ≡ Δ*RPQ* .

*Example: *

State whether the following pair of triangles are congruent. If so, state the triangle congruence and the postulate that is used.

*Solution:*

From the diagram, we can see that

- Δ
*ABC *and Δ*PQR *are right triangles

* AC* = *PQ* (hypotenuse)
* AB* = *PR* (leg)

So, triangle

*ABC* and triangle

*PQR* are congruent by the Hypotenuse Leg Theorem.

**Hypotenuse - Leg Congruence Theorem**
Explains why HL is enough to prove two right triangles are congruent using the Pythagorean Theorem

The following video shows more examples of the Hypotenuse Leg (HL) Theorem and the Angle-Angle-Side (AAS) Theorem

## Prove Triangle Congruence with HL Postulate

HL Postulate (Lesson)

A lesson and proof of the HL (Hypotenuse-Leg) postulate using a two-column proof

#### HL Postulate (Practice)

Practice problems and proofs using the HL (Hypotenuse-Leg) Postulate