Triangle on Beltrami pseudosphere with angle sum $180^\circ$

Question

What characteristic lines on the pseudosphere can form a triangle whose internal angle sum is $180^\circ$?

Answer 1

If by "characteristic lines" you mean "geodesics", then this does not happen. All triangles have angle sum strictly less than $180^\circ$, by the Gauss-Bonnet theorem.

Added: Your comment seems to indicate that you are asking not about "geodesics" but simply about arbitrary smooth curves. If so, then there are almost no restrictions on the angle sum whatsoever. If I start with any triangle having three smooth curves for its sides, meeting at three angles, then I can alter the sides of the triangles by arbitrarily small amounts near each of the three angles ("small" meaning $C^0$) to make the each internal angle be anything in the interval $(0,2\pi)$ and hence their sum can be anything in the interval $(0,6\pi)$.