Euclidean geometry is of great practical value. It has been used by the ancient Greeks through modern society to design buildings, predict the location of moving objects and survey land.
The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry.
The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines:
In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it.
In spherical geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.
