Another Proof

Prove the converse of the Pythagorean Theorem. That is, assume that triangle ABC is given with a^2 + b^2 = c^2, and prove that triangle ABC is a right triangle.

Assume that points A, B, and C form a triangle. Also assume that the triangle is scalene with measures of the sides being in the proportion of a^2 + b^2 = c^2.

AD1 allows for points P, Q, and R. Definition 4.18 allows for a triangle to be formed by P,Q, and R such that it is a right triangle. Let’s assume that two of the sides, a and b, match the lengths of two sides of triangle ABC, and that the third side is q. According to Theorem 5.18, as triangle PQR is a right triangle, then q^2 = a^2 + b^2.

As c^2 = a^2 + b^2 and q^2 = a^2 + b^2 then q^2 = c^2. As all of the sides for triangles ABC and PQR are congruent, Theorem 5.16 states that they are congruent. If they are congruent triangles, then ABC must also be a right triangle.