What is the theorem of finding "x" in this special right triangle?

The only thing that makes it special is that it's simple enough to memorize.

The Pythagorean Theorem says the square of x will be the sum of the squares of the two shorter sides.  But you already know that the sum of the angles is 180 degrees, so the acute angle at the bottom of the diagram is 45 degrees, which implies that the missing SHORT side is also 5 in length.  Therefore 

x^2 = 5^2 + 5^2 =>

x = sqrt(50) = 5*sqrt(2).

So memorize that whenever you have a 45-45-90 triangle, the hypotenuse is sqrt(2) times either of the legs.

x = 5√2        

A right triangle with 45⁰ for the other two angles is an isosceles triangle with sides in the ratio 1 : 1 : √2 (by Pythagoras). Thus in this case, x = 5√2

It is a shortcut of the Pythagorean Theorem stating that the length of the hypotenuse is the square root of 2 times the length of either leg.  5√2 in this case.