The Exterior Angle Theorem Fails in Spherical Geometry

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.

Spherical geometry has no parallel lines, so something must go wrong with the above argument. The figures below show that the exterior angle DCE is not always larger than the interior angle ABE=FCE.

A small spherical triangle A medium spherical triangle A large spherical triangle

These figures were drawn using Spherical Easel.