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Proposition 16

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. | XI.1
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.

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This proposition is used in the proof of the next proposition as well as proposition XI.24.

Book XI Introduction - Proposition XI.15 - Proposition XI.17.

© 1996
D.E.Joyce
Clark University