parallellism in Euclidean planeParallellismInEuclideanPlane2007-06-05 13:32:042007-08-28 05:29:36DefinitionEuclidean geometry of planeparallelparallel linesparallelismEuclidean geometry% this is the default PlanetMath preamble. as your knowledge
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Two distinct lines in the Euclidean plane are {\em parallel} to each other if and only if they do not intersect, \PMlinkname{i.e.}{Ie} if they have no common point. By convention, a line is parallel to itself.
The {\em parallelism} of $l$ and $m$ is denoted
$$l \parallel m.$$
Parallelism is an equivalence relation on the set of the lines of the plane. Moreover, two nonvertical lines are parallel if and only if they have the same slope. Thus, slope is a natural way of determining the equivalence classes of lines of the plane.