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What kind of triangles can you use with the Pythagorean Theorem?
Right triangles - those with a 90-degree angle.
What kind of triangle it used to describe for pythagorean inequality theorem?
Obtuse
Did anyone oppose to the pythgreom theorm?
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.
Why was Pythagorean Theorem become necessary to incorporate in the number system?
Your question is so confusing that I almost trashed it and am not sure yet what you want to know but I have a possible idea : consider a right triangle each of whose legs have length 1. By the Pythagorean theorem, the hypotenuse has length equal to the square root of 2. The square root of 2 is irrational- one can prove it is not equal to any fraction of integers, yet it is obviously is a number of some kind. Thus the number system had to be extended to include numbers of this kind.
What kind of triangles make up the faces of a triangular hexahedron?
They can be any kind of triangles.
