Converse of Pythagorean Theorem (3.8)

Proof: Consider and assume . Construct a second such that and and is right by construction. Since is a right triangle, we can use the Pythagorean Theorem to determine the length of . So, . Since we know that and , we can see that . This means that by Common Notion 1. Since all of the sides of and are congruent, we can conclude by Proposition 8 that the triangles themselves are congruent so their angles must also be congruent. By construction, we know that is a right angle, so must also be a right angle. Therefore, the converse of the Pythagorean Theorem is true.