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# 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, $m\,||\,l$.

## Mathematics Subject Classification

51M04*no label found*

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