The Pythagorean Theorem states that if is a triangle right angled at , then,
The converse of the Pythagorean Theorem states that if
then triangle is a right triangle right angled at .
This means that in order to prove the converse of the Pythagorean Theorem, we need to prove that in the figure above is a right angle. Now, we discuss the proof.
In a triangle with sides , and (see figure above), if
then is a right triangle with a right angle at .
Let be a triangle such that , and right angled at (see figure below). If we let , since is a right triangle, by the Pythagorean Theorem
But from the supposition,
From (1) and (2)
Since and are both positive (Can you see why?), we can therefore conclude that
This means that length of the three corresponding pairs of sides of triangle and triangle are equal.
Therefore, by SSS Congruence, .
Since and are corresponding angles, degrees.
And hence we have proved that triangle is right angled at
* Note that in the proof, is the opposite of angle , is the opposite of angle , and is the opposite side of angle .