There are more than three types, although 2 main types are Euclidean and Taxicab Geometry
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Euclidean geometry, non euclidean geometry. Plane geometry. Three dimensional geometry to name but a few
Plane Geometry and Solid Geometry
The 2 types of non-Euclidean geometries are hyperbolic geometry and ellptic geometry.
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No. The rules of two dimensional geometry can only be used for two dimensional geometry. You can take the basic principles of two dimensional geometry and alter them slightly to be able to apply to three dimensional solids