Parallel

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Two lines are said to be parallel if they lie in the same plane but do not intersect.

(Note that by the first part of this definition, skew lines are not considered to be parallel.)

One of the postulates (or axioms) of Euclidean geometry is that given a plane, a line on that plane and a point on that plane not on the line, there is exactly one line passing through the point parallel to the given line. This axiom has historically proven to be contentious, with many attempts made from the time of the ancient Greeks onward to prove it from the other axioms. These attempts all failed, and in (YEAR?) it was proven by (PERSON?) that the Parallel Postulate did not follow from the other axioms of Euclidean geometry.

More recently, in the late 19th century it was discovered that negations of the Parallel Postulate led to different, interesting geometric systems.

One example of such a system is spherical geometry. If you and I begin on different longitudes and travel in parallel directions (say, both travel due north), our paths will eventually cross each other (probably at the North Pole). In other words, spherical geometry is one model of a system in which a given line has no parallel lines.