parallel postulate

Etymology
From the reference to parallel lines in the definition as formulated below, following Scottish mathematician ; this wording leads to a convenient basic categorisation of Euclidean and non-Euclidean geometries. The original formulation in makes no mention of parallels.

Noun

 * 1)  An axiom of Euclidean geometry equivalent to the statement that, given a straight line L and a point P not on the line, there exists exactly one straight line parallel to L that passes through P; a variant of this axiom, such that the number of lines parallel to L that pass through P may be zero or more than one.

Usage notes
Equivalent formulations include:
 * If a line segment intersects two straight lines forming two interior angles on the same side that sum to 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.
 * The sum of the angles in any triangle equals a straight angle.
 * Numerous others listed at

Variations can be classified as follows:
 * No straight line exists that is parallel to L and passes through P;
 * Exactly one straight line exists that is parallel to L and passes through P;
 * At least two straight lines exist that are parallel to L and pass through P.

It is also possible to forego the postulate entirely, as is the case in.

Synonyms

 * Euclid's fifth postulate,

Translations

 * Finnish: