It's the bottom of a 3-D shape or solid. For example, a square-based prism. The square face would be the base. Here's another example, a triangluar-based pyramid. The triangle would be the base.
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a face
If the tent has a square base, it is a pyramid.
postulates cannot be proved, they are the base of geometry and there isn't anything to prove it with. if the postulates were wrong then all of euclidian geometry would be wrong. that is like saying how do we know the English language is correct, it is the basis for communication and if it wasn't, then how would speaking the language work?
Euclidean geometry, non euclidean geometry. Plane geometry. Three dimensional geometry to name but a few
There are different kinds of geometry including elementary geometry, Euclidean geometry, and Elliptic Geometry.