Axiom:Euclid's Fifth Postulate

If two lines are drawn which intersect a third in such a way that the sum of the measures of the two interior angles on one side is less than the sum of the measures of two right angles, then the two lines must intersect each other on that side if extended far enough.

This is equivalent to the Parallel Postulate.

Many mathematicians attempted to prove this postulate, as it seems less intuitive than the rest of Euclid's Postulates. Euclid himself avoided using this postulate until the 29th Proposition in his seminal The Elements. Eventually, in 1823 Lobachevsky and Bolyai independently realized that self-consistent, non-euclidean geometries could be developed by not accepting this postulate. Therefore, the postulate is in fact axiomatic for Euclidean Geometry.