The two commonly mentioned non-Euclidean geometries are hyperbolic geometry and ellptic geometry.
If one takes "non-Euclidean geometry" to mean a geometry satisfying all of Euclid's postulates...
Non-Euclidean geometries are based on the negation of his parallel postulate (his fifth postulate). The other Euclidean postulates remain.A rephrasing of Euclid's parallel postulate is as follows:For...