I have just done some fairly common exercises around geodesics in the hyperbolic half-plane. In particular, I showed that vertical lines and semicircles form geodesics.
If I take a vertical line geodesic, and a point not on it and sufficiently far from the $x$-Axis (to prevent semicircle-geodesics from reaching it), then there is exactly one other geodesic, the vertical line through that point, that doesn't intersect the original geodesic.
Doesn't this violate the fifth postulate that generates hyperbolic geometry from absolute geometry? Don't I need to be able to find two or more geodesics?
Semicircle-geodesics don't have this problem, there are infinitely many of those.
There are other geodesics. Suppose the vertical line geodesic is given by $x = a$ and the point by $(b, c)$, with $b \ne a$. Choose $d \ne b$. Then there is a geodesic semicircle with endpoint $(d, 0)$ and passing through $(b, c).$ To find it, note that if the center of the semicircle is $(e, 0)$, then we must have $$(e - d)^2 = (e - b)^2 + c^2.$$ Solving this equation, we obtain $$e = \frac{b^2 + c^2 - d^2}{2(b - d)}.$$ Now, if we further suppose that $d$ is between $a$ and $b$, then the semicircle will not intersect the vertical line.