Parallel postulate of euclid

Question

How euclid's 5th postulate transforms in non euclidean geometry? How to visualize that there are more than one lines that are parallel to the point?

Answer 1

To answer your first question, when the parallel postulate is no longer assumed to be true, other geometries such as hyperbolic and ecliptic geometry are created. In hyperbolic geometry, there are infinitely many intersecting lines that do not intersect, but in ecliptic geometry, every line eventually intersects.

What I think is the most important question is your second question,

Playfair's axiom which is a corollary of the parallel postulate says "given a line and a point not on it (the line), then there is only one line that passes through the point AND is parallel to the line", which directly contradicts your statement.

When the original form of Euclid's parallel postulate states that if the angle between two straight lines ($A$ and $B$) and an intersecting line ($C$) is less than $90º$, then those two lines will meet. This implies that when $A$ and $B$ are perpendicular to $C$, then $A$ and $B$ are parallel. We finally get to the corollary by placing the point at the intersection of $A$ or $B$ and $C$, so when the other intersection is moved on the other line, there is only one parallel line passing through it.

I hope this cleared some misconceptions about the exact definitions of the parallel postulate, and what happens in alternate geometries when the parallel postulate is not true.