I was introduced to both the Poincaré disc model and the upper half-plane model and need to prove the statement in the title. Apparently you can do this by showing that sectors in the disc model with central angle < $\pi$ and vertical strips in the upper half-plane model are convex. Using that, you show that a triangle in the upper half-plane is the intersection of three "suitable" convex sets and is therefore also convex.
I understand why vertical strips are convex, but other than that, I'm at a loss.