An axiom is a self-evident statement that is assumed to be true. A theorem is proved to be true.
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There is no difference - synonymous.
Difference between first shifting and second shifting theorem
It is not an axiom, but a theorem.
There is no relationship between slope and the theorem, however the theorem does deal with the relationship between angles and sides of a triangle.
Although the Pythagorean theorem (sums of square of a right angled triangle) is called a theorem it has many mathematical proofs (including the recent proof of Fermats last theorem which tangentially also prooves Pythagorean theorem). In fact Pythagorean theorem is an 'axiom', a kind of 'super law'. It doesn't matter if anyone does oppose it, it is one of the few fundamental truths of the universe.