Verified answer

It is hard to give a complete answer to which geometries don't assume Euclid's parallel postulate, without first specifying precisely what a "geometry" is (ie, what is it that all of the "geometries" must have in common in order to be called "geometries"?) Regardless of how you define it, the examples given on the Wikipedia page below probably meet the definition, and are a good start. [If you get more formal about it--- ie, by explicitly listing which axioms you accept as defining a "geometry"--- there may be more than just the possibilities listed there.]

Just speaking loosely, perhaps the simplest example of something that resembles "geometry" without the parallel postulate holding, is the geometry of the surface of a sphere, with "point" defined to be the usual thing--- a point on the surface of the sphere--- and "lines" taken to mean so-called "great circles" (circles whose centers are the center of the sphere: these give the shortest paths between points on the surface of the sphere, so they are a natural replacement for lines in a curved surface that doesn't contain any). This geometry is non-Euclidean because any two "lines" in this geometry intersect so there is no such thing as parallel lines.

As for proving that the sum of the angles in a triangle is 180, really, you have to use the parallel postulate because without it, it's not true. (The page below has a picture of examples on the sphere where it isn't true.) The standard way of doing it is as follows. Let ABC be a triangle. By the parallel postulate there is a unique line through C that is parallel to the line segment AB, so draw that in. By choosing two points on it, label it line EF (with E on the opposite side of AC as B, and F on the opposite site of BC as A). You now use other theorems about parallel lines (that require the parallel postulate!) to deduce that angle CAB is equal to angle ECA, and that angle BCF is equal to angle ABC--- so the sum of the angles in the triangle is equal to the sum of the angles ACE, ACB, BCF, and this is clearly 180.

Without the parallel postulate this proof breaks down because there may be no line EF through C that is parallel to AB. Or there could be *many* of them, and among them it may not be possible to choose one so that both angle CAB = angle ECA and angle BCF = angle ABC hold at the same time.