If two parallel lines intersected in a point that would violate Euclid's 5th postulate and we would no longer be doing "plane geometry."I believe that it was the replacement of this postulate (which had been viewed for centuries as too complex to be a postulate, and as qualitatively different in structure from the other postulates) with the assumption that hey intersected in a point which led to the first non-Euclidean geometry, which I believe is the geometry of the projective sphere:
Establish x-y axes in the plane.
Set a unit'sphere on the origin.
Map each point, p, in the plane to a unique point, s, on the sphere by drawing a straight line from the "north pole", N, of the sphere to p; s is the point where that line intersects the sphere.
Under this mapping all lines eventually intersect at "the point at infinity", which maps into N, the north pole of the sphere. (Visualize the line from N to the origin, then watch as you move p, the point in the plane, along the x-axis. You can "see" s, the point of intersection with the sphere, moving steadily upward. And then realize that it doesn't matter what direction the point is moving in.)
But this is all WITHOUT the 5th postulate, and consequently it requires a perspective from outside the plane (namely, from N).