The converse of the Pythagorean Theorem is:
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
That is, in ΔABC, if c^2=a^2+b^2 then ∠C is a right triangle, ΔPQR being the right angle.
proof;
We can prove this by
contradiction.
Let us assume that c^2=a^2+b^2 in ΔABC and the
triangle is not a right triangle.
Now consider another triangle ΔPQR. We construct ΔPQR so that PR=a, QR=b and ∠R is a right angle.
By the Pythagorean Theorem, (PQ)^2=a^2+b^2.
But we know that a^2+b^2=c^2 and c=AB.
So, (PQ)^2=a^2+b^2=(AB)^2.
That is, (PQ)^2=(AB)^2.
Since PQ and AB are lengths of sides, we can take positive square roots and say:
PQ=AB
That is, all the three sides of ΔPQR are congruent to the three sides of ΔABC.
So, the two triangles are congruent by the Side-Side-Side Congruence Property.
Since ΔABC is congruent to ΔPQR and ΔPQR is a right triangle, ΔABC must also be a right triangle.
This is a contradiction.
Therefore, our assumption must be wrong.