# 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 $ABC$ and $DEF$ be right triangles right angled at $C$ and $F$. Let $AB \cong DE$ and $\angle A \cong \angle D$.

It is given that angle $\angle A \cong \angle D$ and $AB \cong DE$.

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

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

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.