In 300 BC Euclid invented euclidean geometry based on 5 postulates:

1. To draw a straight line from any point to any other.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to each other.

5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

However it was not clear whether this was the minimal set of postulates. In particular the 5th postulate was thought to be derivable. But all attempts to derive it failed, for more than 2 thousand years.

In 1823 Bolyai began to realise that it was in fact an independent postulate that need not be true, and separately Lobachevsky published examples of non euclidean geometries from 1829 onwards, where this 5th postulate was denied.

Mathematicians were not receptive to this concept; as there was no proof of their consistency. Eventually a way of embedding 2 dimensional non-euclidean geometries in normal 3-space was found, and their consistency became obvious.

Therefore in the modern world, mathematicians are very comfortable with these geometries- for example the surface of the earth is non-euclidean due to its approximately spherical shape, and further, according to Einstein's General Theory of Relativity the very space around and in the earth is non-Euclidean, and therefore the total interior angles of a triangle differs from 180 degrees by a tiny amount due to the curvature induced by the gravity of the earth.

In fact, due to the space inside the Earth 'slumping' slightly, it is thought that the distance between the poles from the outside is lower than the distance if you were to drill a hole down through the Earth and measure the distance that way.