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Parallel postulate
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<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the parallel postulate is the fifth postulate in <a href="/facts/Euclid%2527s_Elements/nPKvCIxs">Euclid's <i>Elements</i></a> and a distinctive <a href="/facts/Axiom/mLjP14Mc">axiom</a> in <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>. It states that, in two-dimensional geometry: </p> <blockquote> <p>If a <a href="/facts/Line_segment/V8E0M4qk">line segment</a> intersects two straight <a href="/facts/Line_(mathematics)/gJqsr4Gj">lines</a> forming two interior angles on the same side that are less than two <a href="/facts/Right_angle/psaslRq7">right angles</a>, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. </p> </blockquote> <p>This postulate does not specifically talk about parallel lines; it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates. </p><p><i>Euclidean geometry</i> is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. </p><p>The postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually, it was discovered that inverting the postulate gave valid, albeit different geometries. A geometry where the parallel postulate does not hold is known as a <a href="/facts/Non-Euclidean_geometry/efKa7De2">non-Euclidean geometry</a>. Geometry that is <a href="/facts/Logically_independent/xW9tCrGB">independent</a> of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as <a href="/facts/Absolute_geometry/ABKfjdn2">absolute geometry</a> (or sometimes "neutral geometry"). </p>