Parallelpostulate

The parallel postulate, traditionally called Euclid's fifth postulate, states that through a given point not on a line there passes exactly one line that does not meet the given line, i.e., a line parallel to the given line. In alternate wording used in many treatments, if a straight line intersects two straight lines and the interior angles on the same side sum to less than two right angles, the two lines meet on that side; if they sum to exactly two right angles, the lines are parallel.

Historically, Euclid included the postulate as an axiom after the first four seemingly self-evident postulates. He proved numerous theorems using the assumption, but the parallel postulate was controversial, and many mathematicians attempted to prove it from the other axioms. These efforts failed, revealing the postulate’s independence from the first four.

The realization of independence led to the development of non-Euclidean geometries in the 19th century. In hyperbolic geometry, through a point not on a line there are infinitely many lines that do not meet the given line. In elliptic (spherical) geometry, no line is parallel to another. These geometries are consistent with suitably altered axiom systems and broaden the scope of geometric thought beyond Euclidean assumptions.

In modern treatments, an equivalent form is Playfair's axiom: through a given point not on a line there is a unique parallel line. Many textbooks adopt the parallel postulate in this or an equivalent form to characterize Euclidean plane geometry. The postulate is independent of the other axioms, and it underpins the standard theorems of Euclidean geometry.