# parallelism of line and plane

Parallelity of a line and a plane means that the angle between line and plane is 0, i.e. the line and the plane have either no or infinitely many common points.

Theorem 1. If a line ($l$) is parallel^{} to a line ($m$) contained in a plane ($\pi $), then it is parallel to the plane or is contained in the plane.

Proof.
So, $l||m\subset \pi $. If $l\not\subset \pi $, we can set a set along the parallel lines $l$ and $m$ another plane $\varrho $. The common points of $\pi $ and $\varrho $ are on the intersection^{} line $m$ of the planes. If $l$ would intersect the plane $\pi $, then it would intersect also the line $m$, contrary to the assumption^{}. Thus $l||\pi $.

Theorem 2. If a plane is set along a line ($l$) which is parallel to another plane ($\pi $), then the intersection line ($m$) of the planes is parallel to the first-mentioned line.

Proof. The lines $l$ and $m$ are in a same plane, and they cannot intersect each other since otherwise $l$ would intersect the plane $\pi $ which would contradict the assumption. Accordingly, $m||l$.

Title | parallelism of line and plane |
---|---|

Canonical name | ParallelismOfLineAndPlane |

Date of creation | 2013-03-22 18:47:58 |

Last modified on | 2013-03-22 18:47:58 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 51M04 |

Related topic | ParallelismOfTwoPlanes |