Suppose you had this triangle, which is not necessarily a right triangle:



You only know that  but you don't know necessarily 
that angle  is 90°.

Now you construct another triangle, this time a RIGHT triangle 
with the SAME SIZE legs,  and , but with its hypotenuse
 not given to be equal to :



But we know that the Pythagorean theorem DOES hold true for triangle
DEF because we constructed it so that it is a right triangle with
angle E being a right angle. Therefore



And we are given that



So  because both are equal to .

and we can take positive square roots of both sides:



or 

Now we have all three sides of triangle  equaling 
all three corresponding sides of triangle , so the
two triangles are congruent, and therefore angle 
equals angle E which equals 90°, for they are corresponding
parts of congruent triangles. Therefore  is a
right triangle.

Therefore all we need know is that  to be
able to conclude that triangle  is a right triangle.

So the only time the Pythagorean theorem works is when we
have a right triangle.

Edwin