5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right
angles, then the two lines inevitably must intersect
each other on that side if extended far enough. This postulate is equivalent to what
is known as the parallel postulate.

Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute
geometry") for the first 28 propositions of the Elements,
but was forced to invoke the parallel postulate
on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized
that entirely self-consistent "non-Euclidean
geometries" could be created in which the parallel postulate did not
hold. (Gauss had also discovered but suppressed the existence of non-Euclidean
geometries.)