Let $l_{1}$ and $l_{2}$ be ultraparallel geodesics in the hyperbolic plane. Prove that the $l_{i}$ have a unique common perpendicular.

Suppose now $l_{1}, l_{2}, l_{3}$ are pairwise ultraparallel geodesics in the hyperbolic plane. Can the three common perpendiculars be pairwise disjoint? Must they be pairwise disjoint? Briefly justify your answers.