Question

Describe a spherical triangle in spherical geometry that is a counterexample to theorem 1.

Explain how the spherical triangle proves theorem 1 is false in spherical geometry.

theorem 1

theorem 1 (the exterior angle inequality theorem): An exterior angle of a triangle is greater than each of the remote/nonadjacent interior angles of the triangle.

Answer 1

A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar triangle, and is sometimes called an Euler triangle.

The Exterior Angle Theorem in neutral geometry says that any exterior angle of a triangle is always strictly larger than either nonadjacent interior angle. This theorem plays a crucial role in the proof that there are parallel lines in neutral geometry.

However, spherical geometry has no parallel lines, so something must go wrong with the above argument. The figures below show that the exterior is not always larger than the interior.