If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the base greater than the base. If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another. To see what can go wrong, let us recall Euclid's argument, and see what happens if we try to apply it to triangles drawn on the surface of a sphere. This works perfectly well on the surface of a sphere. This works pretty well on the surface of a sphere.
Vertical angles are equal on a sphere.
This is where the proof would get into trouble on the surface of a sphere. Indeed, the observation that F lies in the interior of angle ECD is tantamount to assuming what it is we are trying to prove. Indeed, on the sphere, the Exterior Angle Theorem and most of its consequences break down utterly. In order to clarify where matters stand on the plane, we will need to look more closely at how our axioms for plane geometry rule out the behavior we have seen on the sphere. Connect C to D. Suppose that angles CAD and ACB were equal.