# Hypotenuse Leg Theorem

Proving Congruent Right Triangles

The hypotenuse leg theorem states that any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles.

### Examples

##### Example 1
Given:
$\overline{AC} \cong \overline{ZY} \\ \overline{AB} \cong \overline{XZ} \\ m\angle ACB = m\angle ZYX = 90^{o}$
Prove:
$\triangle ACB \cong \triangle ZYX$
Statement Reason
$$\angle ACB$$ and $$\angle ZYX$$ are both right angles Given
$$\triangle ABC$$ and $$\triangle XYZ$$ are right triangles Right Triangles have a right angle
$$\overline{AB} \cong \overline{XZ}$$ Given
$$\overline{AC} \cong \overline{ZY}$$ Given
$$\triangle ACB \cong \triangle ZYX$$ Hypotenuse Leg Theorem
##### Example 2

The following proof simply shows that it does not matter which of the two (corresponding) legs in the two right triangles are congruent.

• ABC and XZY are right triangles since they both have a right angle
• AB = XZ (hypotenuse) reason: given
• CB = XY (leg) reason: given
• ABC XYZ by the hypotenuse leg theorem which states that two right triangles are congruent if their hypotenuses are congruent and a corresponding leg is congruent.