Converse Pythagorean Theorem

Consider a triangle ABC with side lengths a, b, and c such that c^2 = a^2 + b^2. The idea is to prove that m∠ C = 90^(∘).

Now, construct a right triangle PQR such that PR=b, QR=a, and m∠ R = 90^(∘).

Write PQ in terms of a and b by applying the Pythagorean Theorem. PQ^2 = a^2 + b^2 Remember that c^2=a^2+b^2. Then, set the left-hand sides of these equations equal to each other. c^2 = PQ^2 ⇒ sqrt(c^2) &= sqrt(PQ^2) c &= PQ The last equation implies that the sides of △ ABC are congruent to the sides of △ PQR. a &= QR ⇒ & BC&≅QR b &= PR ⇒ & AC&≅PR c &= PQ ⇒ & AB&≅PQ Triangles ABC and PQR are congruent by the Side-Side-Side Congruence Theorem. This implies that ∠ C≅ ∠ R, because corresponding angles are congruent. Remember, by construction m∠ R=90^(∘). Therefore, m∠ C=90^(∘) which completes the proof.