TITLE: Difference between axioms, theorems, postulates, corollaries, and hypotheses
QUESTION [66 upvotes]: I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)

REPLY [2 votes]: Axiom: Not proven and known to be unprovable using other axioms
Postulate: Not proven but not known if it can be proven from axioms (and theorems derived only from axioms)
Theorem: Proved using axioms and postulates
For example -- the parallel postulate of Euclid was used unproven but for many millennia a proof was thought to exist for it in terms of other axioms. Later is was definitively shown that it could not (by e.g. showing consistent other geometries). At that point it could be converted to  axiom status for the Euclidean geometric system.
I think everything being marked as postulates is a bit of a disservice, but also reflect it would be almost impossible to track if any nontrivial theorem does not somewhere depend on a postulate rather than an axiom, also, standards for what constitutes 'proof' changes over time. 
But I do think the triple structure is helpful for teaching beginning students. E.g. you can prove congruence of triangles via SSS with some axioms but it can be damnably hard and confusing/circular/nit-picky, so it makes sense to teach it as a postulate at first, use it, and then come back and show a proof.