Riemannian, Lobachevskian , or Euclidean geometry - it's all there ....
It was the "biggest" problem in Geometry almost since the time of Eucleides who sattled the axioms (like: through two distinct points, there is one and just one line passing etc.) +/-300 before Christ. For me, this was solved by Riemann (formally by David Hilbert). If we admit the fifth axiom of Euclides, we can prove that the sum is just 180 for each triangle. If we do not admit this and nothing equivalent to it, the answer is "any", but not 180 (simplification - I think, compactness?).
The problem was: "Is it possible to derive the fifth axiom from the 4 (preceeding ones) remaining ones?" If it were, then (without assuming the fifht one), each triangle has 180 as sum of angles. (This is may be a bit academic and just recalling what I wrote above, but "Why not to ask such questions?".) The names then were Saccheri, who wanted to prove it and who was constructing 'worlds', where one can not prove the 5. axioms, i.e. worlds where the axiom was not true. Then Lobatchevsky, Bolay - should be mentioned! (the younger) and then the Riemann and Hilbert. Very interesting is the role of Gauss in these discoveries. Very nice are the books of the Czech mathematician Petr Vopenka (Fourth discussion with geometry), where one can find ( a speculation-?, but actually some hints say that is true - e.g., the Riemann's Habilitation Lecture "Uber die Hypothesen welche der Geometrie zugrunde liegen." which was chosen by Gauss. This shows that Guass was at least interested in this...) and gthere is a historical evidence that Gauss was awere of the existence of worlds where the fifth axiom is not true (as our wordl according to current and any next? Physics). In English literature, one can learn a lot of basic geometry and differential geometry and also a connection to the 5. axiom and Saccheri work The Geometry from differential viewpoint. Sorry for not mentioning any English mathematican connected to the fifth axiom and any other nationalities, actually -I would be glad if somebody can give some other names.
Actually, in a non-Euclidean situation, the sum is not necessarily 180 degrees.
Euclidean and Non-Euclidean Geometry
Euclidean: the sum of the angles of any triangle is always equal to 180°
Hyperbolic: the sum of the angles of any triangle is always less than 180°
Elliptic: the sum of the angles of any triangle is always greater than 180°; geometry in a sphere with great circles.
For example, the sum of the interior angles of a Poincare triangle is always less than 180 deg. See attached a sample Poincare triangle in the image.
Yes, depending on the kind of geometry one is considering. This gives the following types of geometries. Euclidean geometry(ie angle sum = 180 deg), hyperbolic geometry (ie. angle sum < 180 deg) and spherical geometry (ie. angle sum > 180 deg).
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