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.

hypotenuse angle theorem

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.

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