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Can you please describe the geometry in which the sum of the angles of the triangle can be less than 180 degrees?

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Hyperbolic geometry, obtained by taking Euclid's axioms and changing the parallel postulate to its converse, will have interior angles which are smaller than the Euclidean ones. How much smaller depends on the size of the object, for hyperbolic geometry has an absolute length scale which relates lengths and angles. You can even have ideal triangles with angle sum zero. There are many models of hyperbolic geometry, but the ones where angles are most apparent are those by Poincaré: his disk and half plane models.

Also note that any surface of negative Gaussian curvature will result in angles smaller than the Euclidean ones. So for limited areas you can visualize this on a saddle-shaped surface.

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