Parallel (geometry)

Key characteristics

Parallelism in Euclidean geometry has several elementary properties that are useful in proofs and applications. When two non-vertical lines in the plane are parallel, they have equal slopes. In linear-algebra terms, direction vectors of parallel lines are scalar multiples of one another. For two lines given by ax + by + c1 = 0 and ax + by + c2 = 0, the perpendicular distance between them equals |c2 - c1| / sqrt(a^2 + b^2). If a line intersects one of two parallel lines, corresponding and alternate interior angles formed by a transversal are equal; conversely, these equal-angle conditions imply parallelism.

History and axioms

Parallelism has been central to geometry since antiquity. Euclid's Elements treated the behavior of parallel lines in a set of postulates and propositions; the so-called parallel postulate was long regarded as less self-evident than the others. In the 19th century alternative formulations such as Playfair's axiom ("through a point not on a line there is exactly one line parallel to the given line") became standard. The study of consistency and alternatives to the parallel postulate led directly to the development of non-Euclidean geometries.

Uses and examples

• Practical design: architecture, engineering drawings, and machine parts use parallel lines and planes to ensure proper alignment and fit.
• Computer graphics: parallel projection preserves parallelism of lines and is used for technical illustrations and isometric views.
• Mathematical proofs: parallelism yields angle equalities and congruent triangles used in many geometric arguments.
• Analytic geometry: testing parallelism via slopes or vector cross products is common in coordinate methods.

Related concepts and notable distinctions

In three-dimensional space, "parallel" may refer to parallel lines or parallel planes; two lines in 3D that do not intersect need not be parallel — they can be skew, meaning they lie in different, non-parallel planes and have no point of intersection. In spherical geometry, true parallel lines do not exist because great circles always meet; in hyperbolic geometry there are infinitely many distinct lines through a point that do not intersect a given line, illustrating how parallelism depends on the underlying geometry. For further reading see parallel lines and related axioms axiomatics.