Parallel postulate

Parallel Postulate

Also called Euclid's Fith Postulate on account of it being the fifth postulate in Euclid's Elements.

If a line segment intersects two straight lines forming two interior angles on the same side sum to less than two right angles then the two lines segments, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

The Parallel Postulate is equivalent to Playfair's Axiom.

The Parallel Postulate is the only Postulate of Euclidean Geometry which fails for Non-Euclidean Geometry.

Many attempts were made to prove the "parallel postulate" in terms of Euclid's first four postulates. The independent discovery of Non-Euclidean Spaces by Gauss and Lobachevsky finally demonstrated the necessity of postulate.