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The axioms are not postulates.
euclidean Geometry where the parallel line postulate exists. and the is also eliptic geometry where the parallel line postulate does not exist.
Euclidean geometry is the traditional geometry: it is the geometry of a plane surface, as developed by Euclid. Among other things, it is based on Euclid's parallel postulate which said (in effect) that given a line and a point outside that line there could only be one line through that point that was parallel to the given line. It has since been discovered that both alternatives to that postulate - that there are many such lines possible and that there are none - give rise to consistent geometries. These are non-Euclidean geometries.
One main characteristic of non-Euclidean geometry is hyperbolic geometry. The other is elliptic geometry. Non-Euclidean geometry is still closely related to Euclidean geometry.
The geometry of similarity in the Euclidean plane or Euclidean space.