Definition:Parallel (Geometry)

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This page is about Parallel in the context of Geometry. For other uses, see Parallel.

Definition

Lines

In the words of Euclid:

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction.

(The Elements: Book $\text{I}$: Definition $23$)


The contemporary interpretation of the concept of parallelism declares that a straight line is parallel to itself.


Planes

Two planes are parallel if and only if, when produced indefinitely, do not intersect at any point.


In the words of Euclid:

Parallel planes are those which do not meet.

(The Elements: Book $\text{XI}$: Definition $8$)


The contemporary interpretation of the concept of parallelism declares that a plane is parallel to itself.


Line Parallel to Plane

Let $L$ be a straight line.

Let $P$ be a plane.

Then $L$ and $P$ are parallel if and only if, when produced indefinitely, they do not intersect at any point.


Sources