What exactly is an axiom, and can an axiom be proven? I've come here after reading about proving SAS congruence, where the response I read was about how it was an axiom. 
I'm not sure, however, what exactly is an axiom. And can it be proven? 
And if it can't be proven, how can one just assume it true? 
 A: What exactly is this question, and can this question be answered?
A: We hold these truths to be self-evident...axioms are a set of basic assumptions from which the rest of the field follows.  Ideally axioms are obvious and few in number.
An axiom cannot be proven.  If it could then we would call it a theorem.  However, there may be two concepts that are equivalent.  And we might state one as an axiom and the other as a theorem.  But, we would be able to change that priority if we thought it would me more concise.
As for SAS... Euclid proposed 5 "postulates" and 5 "common notions," and the "SAS axiom" is neither.  
Early in "The Elements," Euclid runs into a problem proving congruence.  His solution is to introduce the concept of "superposition."  Euclid is not explicit in what he means, but the suggestion seems to be we could "pick up" one triangle and move it and place the two on top of another, if there is an exact correspondence between points and line segments, the triangles are congruent.  
Today we might call this the group of rigid motions.  But group theory wouldn't be discovered for another 1800 years.
Is this allowed by the axioms?  Not really.  The easy solution is to make an additional axiom.
