In hyperbolic geometry, the sum of the angles in a triangle is?

• A. Two right angles
• B. Greater than two right angles
• C. Less than two right angles
• D. Equal to one right angle

Answer: (C)

In hyperbolic geometry, triangles have interior angles that add up to less than two right angles. This happens because the space has negative curvature, causing geodesics to diverge more than they do in a flat plane. As a result, the angles don’t have to sum to 180 degrees to enclose the same area—there’s an angular deficit. The larger the triangle (and the stronger the curvature in the area), the bigger this deficit becomes. So the sum of the angles is always strictly less than 180 degrees.