# 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 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 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.

### Surfaces

Let $S_1$ and $S_2$ be surfaces in ordinary space.

Let $S_1$ and $S_2$ have the property that:

for every point $P$ on $S_1$, a normal vector passing through $P$ is also a normal vector to $S_2$

and:

for every point $Q$ on $S_2$, a normal vector passing through $Q$ is also a normal vector to $S_1$.

Then $S_1$ and $S_2$ are parallel.

## Also see

• Results about parallel can be found here.