**The Parallel Postulate**: Given a line, and a point not on that line, there is one and only line that 1) passes through that point and 2) is parallel to the other line.

This is a foundation of Euclidean geometry, Euclid's fifth postulate (Actually I find it's a version of Playfair's Axiom, which is equivalent). It is, however, not necessarily so; coherent geometrical systems have been constructed under alternate assumptions, e.g.:

*Hyperbolic geometry*, or Lobachevskian geometry (or even Lobachevsky-Bolyai-Gauss geometry), says that there may be *many* lines that pass through our point and do not intersect our line.

*Elliptic geometry*, or Riemannian geometry, says that there are *no* such lines.

An example of the differences these systems may make in the "real world": Euclidean triangles have 180 degrees, Riemannian triangles have more than 180 degrees, and Lobachevskian triangles have less than 180 degrees, the difference depending on the size of the triangle.