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Concept
Parallel (geometry)
Summary
In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines.
Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry.
In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism.
The parallel symbol is . For example, indicates that line AB is parallel to line CD.
In the Unicode character set, the "parallel" and "not parallel" signs have codepoints U+2225 (∥) and U+2226 (∦), respectively. In addition, U+22D5 (⋕) represents the relation "equal and parallel to".
The same symbol is used for a binary function in electrical engineering (the parallel operator). It is distinct from the double-vertical-line brackets, U+216 (‖), that indicate a norm (e.g. as well as from the logical or operator (||) in several programming languages.
Given parallel straight lines l and m in Euclidean space, the following properties are equivalent:
Every point on line m is located at exactly the same (minimum) distance from line l (equidistant lines).
Line m is in the same plane as line l but does not intersect l (recall that lines extend to infinity in either direction).
When lines m and l are both intersected by a third straight line (a transversal) in the same plane, the corresponding angles of intersection with the transversal are congruent.
Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, and so, are "more complicated" than the second.
Official source
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