For any three points in a spacetime of constant positive (constant negative) curvature plausibly regarded as vertices of a triangle, is it always the case that the interior angles of the resulting triangle sum to greater than (less than) 180 degrees? I thought the answer to this question was straightforwardly 'yes'. But, given the following reason, I'm no longer sure about this. It was recently pointed out to me that, with respect to flat (i.e., Minkowski) spacetime, only triangles whose vertices are spacelike separated have interior angles summing to 180 degrees, and when they aren't so separated, the angles are meaningless.
So I'm wondering if the same situation arises in spacetimes of constant positive (constant negative) curvature. That is, in such spacetimes is it also the case that the interior angles of triangles sum to greater than (less than) 180 degrees only when the points of the vertices are spacelike separated, and when they aren't so separated, the angles are meaningless?