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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. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry ...
Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive.
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): In a plane , given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.
Parallel (geometry), two lines in the Euclidean plane which never intersect Parallel (operator) , mathematical operation named after the composition of electrical resistance in parallel circuits Science and engineering
The distance between two parallel lines in the plane is the minimum distance between any two points. Formula and proof. Because the lines are parallel, the ...
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting" [1] [2]) the metric notions of distance and angle. As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines.