The fifth postulate is one of the most famous statements Euclid proposes.  It states that if a straight line intersects two other straight lines, and the sum of the interior angles on one side of the transversal is less than two right angles (180°), then the two lines being intersected will eventually intersect themselves on that side. In hyperbolic geometry, this statement does not hold. Here, parallel lines are defined to be lines that, when extended so that they have infinite length, don’t intersect. What made mathematicians so interested in this postulate is the fact that it is significantly more complicated than any of the other postulates, and sounds like a theorem that can be proven, rather than something that can be assumed to be true without proof. For millennia, people have tried to prove the fifth postulate from only the first four, but it turns out that Euclid was right; the fifth postulate was only a choice. This is actually what motivated the creation of hyperbolic geometry; hyperbolic geometry verifies that the fifth postulate is indeed a postulate. 

 

One equivalent statement to the fifth postulate is that given a line and a point not on it, there’s always one and only one line you can draw that passes through the given point and is parallel to the given line. In hyperbolic geometry, there are infinitely many of these lines you can draw, because hyperbolic lines have a tendency to diverge.