In the current semester I've taken a course about Non-Euclidean Geometry. During the course, we presented two types of non-euclidean geometries: the spherical geometry and the hyperbolic geometry. So that makes three distinct geometries, along with the familiar euclidean one.
However, it seems we weren't really taught what exactly a geometry is - what are our expectations from something we'd like to call "a geometry"?
If I were asked to answer this question, I would say that a geometry must have notions of point, line, angle, distance, length and area.
But in another course- Differential Geometry - we've learned about riemannian metrics on open subsets of the plane, which endow the set with all the mentioned notions (distance being the infimum of lengths of curves connecting two points, lines being geodesics...).
So, if I take an arbitrary riemannian metric, and look at the geometric notions it induces, what deprives this structure from being called a geometry? What is so special about the three that we've been introduced?
I'd be happy if you try to include intuition and motivation as a part of your answers. Thanks!