Write a paragraph proof about a theorem on right triangle

Question

Hypotenuse angle theorem, if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.

Answer 1

The right angles are congruent because the triangles are given as right triangles and all right angles are congruent. One pair of acute angles is congruent (given). The hypotenuses are congruent given. At this point it has been shown that two pairs of angles are congruent and a non-included side. That means the two triangles are congruent by AAS.

If AAS is not available at this point in your course, you can show that the other pair of acute angles is also congruent--acute angles of a right triangle are complementary, and complements of congruent angles are also congruent. At this point the triangles are congruent by ASA (two angles and an included side--the hypotenuse).

This HA "theorem" could also be called a corollary (a statement easily proved based on previously proved theorem or accepted postulate).