Proof: Converse of Pythagorean Theorem
Proof: Converse of Pythagorean Theorem
Converse of the Pythagorean Theorem: Suppose a, b, and c are the lengths of the sides of ABC. If , then C is a right angle.
Proof:
Assume ABC is a triangle for which . We wish to prove that C is a right angle.
Construct a triangle XYZ such that Z = 90, BC = YZ, CA = ZX. In XYZ, since Z = 90, we know that by the Pythagorean Theorem. By construction, we can also say that . By our assumption, we know that . Thus, we can conclude that . Moreover, we can note that YX = BA. By SSS criterion, we can conclude that ABC XYZ. Therefore, C = Z meaning C is a right angle.