TITLE: Side-Angle-Side Congruence and the Parallel Postulate
QUESTION [8 upvotes]: Is there a link between the side-angle-side congruence of triangles and the parallel postulate? Specifically, does it follow from Euclid's first four axioms alone? In fact, does it even follow from all five?

REPLY [4 votes]: You would probably do better to use either Hilbert's or Tarski's axioms, since Euclid's axioms aren't rigorous. In fact, non-Pasch geometries (see here) arguably satisfy Euclid's axioms. 
In other words, since Euclid had a muddled understanding of Euclidean geometry, attempting to use his axiomatic system without modifications will lead to you having a muddled understanding of Euclidean geometry as well.
They are separate axioms in two rigorous axiomatizations of Euclidean geometry.
In both of the (rigorous) axiomatic systems mentioned above, the SAS postulate (five-segment axiom in Tarski's system and the last congruence axiom in Hilbert's system) and the parallel postulate (axiom 10 of Tarski, and here in Hilbert) are different axioms. 
I don't know how much has been proven for Hilbert's axioms, but since Tarski's axiom system is simpler, most of the axioms have been proven to be independent of one another, including the SAS and the parallel postulate. (See e.g. here.)
There exist valid geometries satisfying SAS but not the parallel postulate.
Both Hilbert's and Tarski's axioms, which include SAS as one of the axioms, can also be used to create axiom systems for neutral geometry (by omitting the parallel postulate) and for hyperbolic geometry (by negating the parallel postulate). Since these axiom systems for hyperbolic geometry, which contain both SAS and the negation of the parallel postulate, are also consistent, this also shows that the parallel postulate is independent of SAS. (This is essentially the same argument made by Kristal Cantwell in another answer.)
There exist valid geometries satisfying the parallel postulate but not SAS.
Finally, in taxicab geometry, the parallel postulate does hold, but SAS doesn't. See here:

...the Taxicab Plane satisfies all of the usual axioms (rules) of the Euclidean plane except one of the congruence axioms... We have two triangles obeying SAS which are not congruent!

Since there exist consistent systems of geometry in which (1) the parallel postulate doesn't hold but SAS does, and (2) the parallel postulate does hold but SAS doesn't, we can conclude with certainty that the two are independent of one another.