Which postulates led to the discovery of non-Euclidean geometry?

Answer 1

Adding to what Anand Mehta said, the negation of that statement has two interpretations.

(i) there are zero lines through that point that are parallel to the given line (this is called Elliptic or Reimannian Geometry)

(ii) there is an infinite number of lines that pass through the point and parallel to a given line (this is called Hyperbolic or Lobachevskian Geometry)

I might add that the study of non-Euclidean Geometries are absolutely fascinating.

Answer 2

Euclid's fifth postulate stated that

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

This postulate is also known as the Parallel postulate which states that

Given a line and a point which is not on that line, there is at most one line parallel to the the given line which is coplanar with the given line and passes through the given point.

Non-Euclidean geometries are obtained from the negation of this postulate.