[b][u]Converse of the Pythagorean Theorem:[/u][/b] Suppose a, b, and c are the lengths of the sides of [math]\Delta[/math]ABC. If [math]a^2+b^2=c^2[/math], then [math]\angle[/math]C is a right angle.[br][br][b][u]Proof:[/u][/b][br]Assume [math]\Delta[/math]ABC is a triangle for which [math]BC^2+CA^2=BA^2[/math]. We wish to prove that [math]\angle[/math]C is a right angle. [br][br]Construct a triangle [math]\Delta[/math]XYZ such that [math]\angle[/math]Z = 90[math]^{\circ}[/math], BC = YZ, CA = ZX. In [math]\Delta[/math]XYZ, since [math]\angle[/math]Z = 90[math]^\circ[/math], we know that [math]YZ^2+ZX^2=YX^2[/math] by the Pythagorean Theorem. By construction, we can also say that [math]BC^2+CA^2=YX^2[/math]. By our assumption, we know that [math]BC^2+CA^2=BA^2[/math]. Thus, we can conclude that [math]YX^2=BA^2[/math]. Moreover, we can note that YX = BA. By SSS criterion, we can conclude that [math]\Delta[/math]ABC [math]\cong[/math][math]\Delta[/math]XYZ. Therefore, [math]\angle[/math]C = [math]\angle[/math]Z meaning [math]\angle[/math]C is a right angle.