The **Pythagorean Theorem** states that if is a triangle right angled at , then,

.

The converse of the Pythagorean Theorem states that if

holds,

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.

**Theorem**

In a triangle with sides , and (see figure above), if

holds,

then is a right triangle with a right angle at .

**Proof**

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

(1).

But from the supposition,

(2).

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 .