parallel postulate

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The axiom of Euclidean geometry which says that, if two straight lines are cut by a transversal and the interior angles on one side add up to less than two right angles, then the two lines meet on that side. It is equivalent to Playfair's axiom, which says that, given a point not on a given line, there is precisely one line through the point parallel to the line. The parallel postulate was shown to be independent of the other axioms of Euclidean geometry in the nineteenth century, when non-Euclidean geometries were discovered in which the other axioms hold but the parallel postulate does not.

Subjects: Mathematics.

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