# The Proof of the Hypotenuse Angle Theorem

Last month, we have discussed the proof of the Hypotenuse Leg Theorem. It states that if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another triangle, then the two triangles are congruent. In this post, we are going to discuss a related theorem on right triangles, the Hypotenuse Angle Theorem or HA Theorem. The HA Theorem states that if the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another triangle, then the two triangles are congruent.

Theorem

If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another triangle, then the two triangles are congruent.

Proof

Let \$latex ABC\$ and \$latex DEF\$ be right triangles right angled at \$latex C\$ and \$latex F\$. Let \$latex AB \cong DE\$ and \$latex \angle A \cong \angle D\$. It is given that angle \$latex \angle A \cong \angle D\$ and \$latex AB \cong DE\$.

Since \$latex \angle C\$ and \$latex \angle F\$ are right angles, \$latex \angle B \cong \angle E\$ (Can you see why?).

So, by ASA Congruence, \$latex \triangle ABC \cong \angle DEF\$.

Therefore, if the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another triangle, then the two triangles are congruent.