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  "body": "\n## Statement\n\nLet two [[lines|0MP9BCF6DP]] in a plane be cut by a third line, the transversal.\nSuppose that the two interior [[angles|0GJAJC6TNZ]] on one side of the transversal have a sum less than two right angles.\nThen the two lines, extended far enough, meet on that side of the transversal.\n\n## Remarks\n\nThis is the fifth postulate of Book I of [[Euclid's Elements|83WHDAV68J]], an axiom and not a theorem.\nOver the axioms of [[absolute geometry|VKFPCSRNDH]], it is equivalent to [[Playfair's axiom|PR7Q3QMEHZ]], and also to the statement that the angle sum of every triangle is two right angles.\nIt is independent of the other axioms: the Euclidean plane satisfies it, and the [[hyperbolic plane|GN4N94GQ0Z]] satisfies all axioms of absolute geometry but not this one.\nMany attempts to prove it from the other postulates, such as those of Saccheri and Lambert, led in the end to the discovery of non-Euclidean geometry.\n\n## Explanations\n\n### L1 human\n\nTake two lines and a third line that crosses both.\nIf the two lines lean toward each other on one side, they meet on that side.\nThis sounds obvious, but it cannot be proved from Euclid's other rules.\n\n### L2 human\n\nThe postulate is the dividing line between Euclidean and hyperbolic geometry.\nResults such as the angle sum $180^\\circ$ of a triangle, the existence of rectangles and the theory of similar triangles depend on it.\nResults of absolute geometry, such as the congruence criteria for triangles and the exterior angle theorem, do not.\nIn proofs one usually uses the equivalent Playfair's axiom, which is easier to apply.\n",
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