How could hyperbolic geometry prove that the sum of the measures of the angles of a trigon is not always 180?

Question

What I mean here is that how will hyperbolic geometry prove that the sum of the angles of a triangle will not be always 180... How could it be since we all know that it's 180 based on Euclidean geometry?

Answer 1

Check differential geometry, which studies, among other things, curved surfaces, where the curvature of every point on it is characterized by the Gaussian curvature, equal to the product of the principal curvatures of the given point. This Gaussian curvature can either be < 0, or = 0, or > 0. As an example of hyperbolic geometry, a "saddle surface" of negative Gaussian curvature everywhere can be found in the link given below. With the very precise mathematical tools of differential geometry, mathematicians have been able to determine that triangles formed by geodestics on such surfaces have angles that add up to less than 180 degrees. That's how "we know".

Also check out links to different topics in differential geometry.

Answer 2

the respond is "extra advantageous than." attempt to think of a triangle on the exterior of a sphere. are you able to visualize how the angles are "stretched" so as that they are extra advantageous than they could be in a Euclidean triangle?