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