Parallel Lines

Dependency: Point–line axiom. A derived definition using only points, lines, incidence, and the condition of having no intersection.

Definition

Let $a$ and $b$ be two lines in the same plane.

If $a$ and $b$ have no common point, or if they coincide, then they are parallel, denoted by $a\parallel b$.

Including coincident lines makes parallelism an equivalence relation: reflexive, symmetric, and transitive. The definition captures “never meet, however far extended” as a precise relation.

Remarks

• “In the same plane” is essential. In space, two lines that neither meet nor lie in one plane are skew lines, not parallel lines.
• Parallel does not mean “corresponding angles are equal” by definition. Equal corresponding or alternate angles are properties and tests proved by the Corresponding/alternate angles ⇔ lines parallel family of theorems. The definition itself concerns common points.
• A parallel through an external point is unique. The existence and uniqueness of a parallel through a point off a line is the theorem Through a point off a line, exactly one parallel exists (Playfair), the Euclidean parallel postulate in this system.