Category:Parallel

This category contains results about Parallel in the context of Geometry.
Definitions specific to this category can be found in Definitions/Parallel.

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.