The parallelism axiom for Euclidean lines states that through any point not on a given line, there passes exactly one line not intersecting the given line. As you can see from the sketch, this statement does not hold in the Poincaré universe: there are at least two lines (blue) through the point not intersecting the given line (green). In fact, there are infinitely many non-intersecting lines: drag either of the blue lines around the point to see them.

You can't drag too far, however: the blue line will eventually touch the green one at an edgepoint, and beyone that, will intersect the green line. We call non-intersecting lines which share an edgepoint asymptotic: otherwise, we call non-intersecting lines divergent. Thus through any point not on a given line, there are two lines asymptotic to the given line and infinitely many divergent from it. (The asymptotic lines are "limit lines" for the divergent ones.) To construct these lines, look at the "Asymptotic line" tool in the Construct folder of the Toolbox.