How do the internal angles of a large triangle sum in an open universe geometry?

Less than $180^{\circ}$

The behavior of the sum of angles within a large triangle serves as a direct test for the local geometry of space. In an open universe, the geometry exhibits negative curvature, often likened to the shape of a saddle or a trumpet flare. In this negatively curved space, the geometry dictates that the sum of the internal angles of any sufficiently large triangle will always be less than $180^{\circ}$. This contrasts sharply with a flat universe where the sum is exactly $180^{\circ}$, and a closed universe where the sum exceeds $180^{\circ}$. This property stems from the fact that in an open geometry, parallel lines diverge, pulling the corners of the triangle apart so their sum decreases relative to flat space.