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 $latex \angle A \cong \angle D$.

hypotenuse angle theorem

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.

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