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Concept
Parallel postulate
Summary
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
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.
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. 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 non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or sometimes "neutral geometry").
Probably the best-known equivalent of Euclid's parallel postulate, contingent on his other postulates, is Playfair's axiom, named after the Scottish mathematician John Playfair, which states:
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.
This axiom by itself is not logically equivalent to the Euclidean parallel postulate since there are geometries in which one is true and the other is not. However, in the presence of the remaining axioms which give Euclidean geometry, each of these can be used to prove the other, so they are equivalent in the context of absolute geometry.
Official source
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