The
parallel postulate (aka the
fifth axiom) doesn't work in spherical geometry, in the most dramatic way: the "
lines" (
geodesics) are exactly the
great circles, or
intersections of
planes going through the centre of the
sphere with the sphere. So
any two great circles belong to two planes sharing a point (the centre of the sphere!). But two planes in
R3 sharing a point must share a
straight line, which intersects the sphere at two
antipodal points. Thus every two great circles intersect at two antipodal points; parallels are simply
impossible here!
Actually spherical geometry just manages to violate another of Euclid's axioms. Between every two points there's a "line" (a shortest path), but it's not always unique! Consider two antipodal points (again). Then there exist infinitely many geodesics between them, and "both ways" on each geodesic have the same length. There's definitely more than just one shortest path between antipodes!