Related Topics: More Geometry Lessons

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

The following diagram shows the Hypotenuse Leg Theorem. Scroll down the page for more examples and solutions of how to use the Hypotenuse Leg Theorem.

### 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

The hypotenuse-leg congruence theorem states that if the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, the two triangles are congruent.

Explains why HL is enough to prove two right triangles are congruent using the Pythagorean Theorem

**Examples of the Hypotenuse Leg (HL) Theorem and the Angle-Angle-Side (AAS) Theorem**
State whether or not the following pairs of triangles must be congruent. If so, state the triangle congruence and name the postulate that is used.

### 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

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