Do the angles in a triangle add up to 180?

Answer 1

Yes, in Euclidean space.

If the sum is not 180° you are not in Euclidean space.

If the three angles of a triangle add up to more than 180° then you are in a spherical space, if the sum is less than 180° it is a hyperbolic space.

Example:

Think of a triangle on the Earth with one corner at the North Pole, the other corners on the Equator 90° apart (say at 0° [west of Nigeria in Africa] the other at 90° W [West of Ecuador in South America]).

Each of these angles is 90° so the sum is 270° -

The surface area of the triangle is one eighth of the surface of the globe. But it is a triangle (on a sphere)!