Non-Euclidean geometry is any geometry that deviates from one or more of
the postulates established in Euclidean geometry. Particularly, Euclid's
parallel postulate is violated in non-Euclidean geometry.

Non-Euclidean geometry refers to geometry that does not satisfy all of the
postulates of Euclidean Geometry. Euclidean geometry is the study of
geometric figures based on modern interpretations of Euclid's postulates: 2
points determine a unique line, the ruler postulates, plane separation,
angle postulates, Side-angle-side postulate, and the parallel postulate (as
defined in SMSG). Any geometry that doesn't follow these postulates would
fall into the category of non-Euclidean geometry.

A non-Euclidean geometry is any geometry that does not satisfy one or more of
the SMSG axioms for geometry.

Non-Euclidean geometry is a category of special geometries that build off
neutral geometry by modifying Euclid's parallel postulate (spherical,
hyperbolic, elliptic).

Non-Euclidean geometry is any geometry that does not follow one or more of
the axioms of Euclidean geometry. Examples of non-Euclidean geometry
include hyperbolic, spherical, and taxicab geometry.

Non-Euclidian geometry is any geometry that breaks any of Euclid's four
postulates or, referencing the SMSG postulates list, breaks any of the
first 16 postulates.

Non-Eucldiean geometry is any geometry that does not exactly follow
all of Euclid's axioms or an equivalent of axioms.