The 2 types of non-Euclidean geometries are hyperbolic geometry and ellptic geometry.
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There are several: hyperbolic, elliptic and projective are three geometries.
Elliptic and Hyperbolic geometry.
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Answer The two commonly mentioned non-Euclidean geometries are hyperbolic geometry and elliptic geometry. If one takes "non-Euclidean geometry" to mean a geometry satisfying all of Euclid's postulates but the parallel postulate, these are the two possible geometries.
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