I would have asked this on a math community but I couldn’t find an active one.
In a spherical geometry, great circles are “straight lines”. As such, a triangle can have two or even three right angles to it.
But what if you go the long way around the back of the sphere? Is that still a triangle?
(Edit:) I guess it’s a triangle! Fair enough; I can’t think of what else you would call it. Thanks, everyone.
It is a triangle. The abstraction of lines in hyperbolic geometry are geodesics and just like three lines form a triangle, so do the geodesics. If you walked along the earth’s surface from the equator to the north pole and back, taking 90 degrees angles every time, you will have felt that you made a triangle by walking straight in three directions.
The reason why the angle sum can be more than 180 degrees is that the sphere has a positve curvature. If you want one with negative curvature and less than 180 degrees angle sum, try to make one on the side of the hole on a torus (look up its curvature if my explanation was lacking).