I know that in Euclidean geometry, the sum of the interior angles of a triangle is exactly $\pi$.
In hyperbolic geometry, I know that the sum of the interior angles of a triangle is $\leq \pi$, and I know that there exist triangles in hyperbolic geometry with interior angles that sum to strictly less than $\pi$.
However, what I don't know is ...
How small can the sum of the interior angles of a triangle in hyperbolic geometry get? Is there a lower bound?
Is there a hyperbolic triangle with interior angle sum equal to, say, $1/10000000$? I suspect it is something nice like the angles have to add up to be at least $\pi/2$ but I don't know if that's true.