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# parallel and perpendicular planes

Theorem 1. If a plane ($\pi$) intersects two parallel planes ($\varrho$, $\sigma$), the intersection lines are parallel.

Proof. The intersection lines cannot have common points, because $\varrho$ and $\sigma$ have no such ones. Since the lines are in a same plane $\pi$, they are parallel.

Theorem 2. If a plane ($\pi$) contains the normal ($n$) of another plane ($\varrho$), the planes are perpendicular to each other.

Proof. Draw in the plane $\varrho$ the line $l$ cutting the intersection line perpendicularly and cutting also $n$. Then $l$ must be perpendicular to $n$ and thus to the whole plane $\pi$ (see the Theorem in the entry normal of plane). Consequently, the right angle formed by the lines $n$ and $l$ is the normal section of the dihedral angle formed by the planes $\pi$ and $\varrho$. Therefore, $\pi\bot\varrho$.

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51M04*no label found*

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