Many times there are problems which are, in a sense "outside of geometry", but are nevertheless amenable to a geometric approach. For example, I may be asked to prove that the ranges of any two affine one dimensional functions must have either one, zero, or infinitely many values in common. An easy proof is to say, "One-dimensional affine functions are lines in Euclidean geometry. Such lines intersect once, no times, or infinitely many times according to Euclid's postulates."
In another case, we may be asked to prove a trigonometric identity or inequality. But instead of appealing to their analytic definitions, we may often draw a circle and some lines and use the theorems of Euclidean geometry.
The theories of linear equations and trigonometry can be developed without bringing in any axioms of Euclid. Typically both theories can be derived from the axioms of set theory. The postulates of Euclid, however, cannot be so derived.
1) Why is Euclidean geometry special in this way? Why won't we be able to use another sort of geometry (hyperbolic?) to such proofs? In other words, Euclidean geometry seems more natural than other sorts - but why?
2) Are these proofs rigorous? Should a geometric argument be just as convincing as a proof in language of the original theory?