Exercise 36 Page 614

By definition, an equiangular triangle has three congruent angles.

Therefore, yes, an equiangular triangle has by definition the same angle measure in spherical geometry. However, you cannot think of an equiangular triangle in spherical geometry in terms of an Euclidean-Geometry-Triangle. If you take an equiangular triangle from Euclidean Geometry, it will not have the same angle measures when placed along a sphere.

The only time spherical geometry and Euclidean geometry can be approximated to be the same is if you zoom in far enough on the surface of a sphere. Then the surface can be approximated with a plane.