Question

what role did the fifth postulate play in the development of non-Euclidean Geometry?

Answer 1

Euclid’s 5th Postulate states that lines will always intersect at some point unless they are parallel. However,this is an axiom, not a theorem. In other words, Euclid just assumed this to be a geometric truth, without proof. Apart from Euclid’s Postulate, there is no guarantee that parallel lines cannot meet.Thus the several varieties of ‘non-Euclidean’ Geometry (where parallel lines can meet) can be entirely consistent.[...]. In this way the euclid fifth postulates play a major role in development of non-euclidean geometry.