Euclid's parallel postulate says:
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.
Spherical geometry is an example of non-Euclidean geometry. The Wikipedia article at https://en.wikipedia.org/wiki/Spherical_geometry#Relation_to_Euclid's_postulates says,
contrary to the fifth (parallel) postulate, there is no point through which a line can be drawn that never intersects a given line.
I think that conclusion skips a step and I want to know how to fill in the steps to draw that conclusion myself.
If we go back to Euclid's parallel postulate it only specifies a sufficient condition for two lines to intersect. The sufficient condition is that two straight lines must intersect a third line such that there must be a pair of interior angles on the same side such that the sum to less than 180°.
But Euclid's parallel postulate does not specify any necessary condition for two lines to intersect. Further it does not specify that there must always exist a line through a point that must be parallel to a given line.
I am aware of Playfair's axiom that states:
In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.
But this is not Euclid's parallel postulate. Even if we show that Playfair's axiom is equivalent to Euclid's parallel postulate, how does the end up showing a contradiction between spherical geometry and Euclid's parallel postulate?
I mean I see no contradiction between spherical geometry and Playfair's axiom either. Playfair's axiom states at most one (not exactly one) line parallel to a given line can be drawn through a point not on the given line. So Playfair's axiom allows no such parallel lines to be drawn as well which is consistent with spherical geometry. Where is the violation of Playfair's axiom in spherical geometry then?
How can we then show in a step-by-step manner that the fact that there is no point through which a line can be drawn that never intersects a given line in spherical geometry contradicts Euclid's parallel postulate?