Answer: No, the triangle sum theorem is not true for a sphere surface.
Explanation:
Consider 3 cities A,B,C at the following locations
The placement of C is important because this then allows angle ABC to be 90 degrees. Angles BAC and BCA are also 90 degrees each because B is directly north of both A and C (all northerly roads lead to the north pole at point B). This bizarre triangle has all three angles of 90 degrees. The angles here sum to 90+90+90 = 270 degrees.
