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law of congruency
There are several. The question will need to be more specific.
The Liouville theorem states that every bounded entire function must be constant and the consequences of which are that it proves the fundamental proof of Algebra.
The Pythagorean Theorem applies only to right triangles. (But they don't prove it.)
It does not.If you consider a right angled triangle with minor sides of length 1 unit each, then the Pythagorean theorem shows the third side (the hypotenuse) is sqrt(2) units in length. So the theorem proves that a side of such a length does exist. However, it does not prove that the answer is irrational. The same applies for some other irrational numbers.