Parallellines

Parallellines, commonly called parallel lines, are two distinct lines in a plane that never meet, no matter how far they are extended. In Euclidean geometry, parallelism is a relation of direction: the lines have the same direction and lie in the same plane. Consequently, parallel lines do not intersect and remain at a constant distance from each other.

In two-dimensional coordinates, parallelism can be expressed by equal slopes. Lines y = mx + b1 and y = mx + b2 with b1 ≠ b2 are parallel. More generally, two nonvertical lines have the same slope, and two vertical lines are parallel if they share x = constant. When two parallel lines are cut by a transversal, corresponding angles are equal and alternate interior angles are equal; consecutive interior angles are supplementary.

In three-dimensional space, two lines are parallel if they are coplanar and do not intersect; if they are not in the same plane, they are skew and not considered parallel. The distance between parallel lines is constant and can be measured along a perpendicular segment joining the lines.

Notationally, parallel lines are denoted by the symbol ∥, as in l1 ∥ l2. The concept underpins much of plane geometry and is a consequence of Euclid's parallel postulate. Parallelism finds applications in architecture, surveying, computer graphics, and physics, wherever consistent directional alignment is required.