If two parallel planes are cut by any plane, then their intersections are parallel.

Let the two parallel planes *AB* and *CD* be cut by the plane *EFGH,* and let *EF* and *GH* be their intersections.

I say that *EF* is parallel to *GH.*

If not, then *EF* and *GH* will, when produced, meet either in the direction of *F* and *H* or in the direction of *E* and *G.*

First, let them meet when produced in the direction of *F* and *H* at *K.*

Now, since *EFK* lies in the plane *AB,* therefore all the points on *EFK* also lie in the plane *AB.* But *K* is one of the points on the straight line *EFK,* therefore *K* lies in the plane *AB.* For the same reason *K* also lies in the plane *CD.* Therefore the planes *AB* and *CD* will meet when produced.

But they do not meet, because, by hypothesis, they are parallel. Therefore the straight lines *EF* and *GH* do not meet when produced in the direction of *F* and *H.*

Similarly we can prove that neither do the straight lines *EF* and *GH* meet when produced in the direction of *E* and *G.*

But straight lines which do not meet in either direction are parallel. Therefore *EF* is parallel to *GH.*

Therefore, *if two parallel planes are cut by any plane, then their intersections are parallel.*

Q.E.D.