Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/261

Rh But planes to which the same straight line is perpendicular are parallel to one another; [XI. 14. therefore the plane passing through AB, BC is parallel to the plane passing through DE, EF.

Wherefore, if two straight lines &c.

PROPOSITION 16. THEOREM.

If two parallel planes be cut by another plane, their common sections with it are parallel.

Let the parallel planes AB, CD be cut by the plane EFHG, and let their common sections with it be EF, GH: EF shall be parallel to GH.

For if not, EF and GH, being produced, will meet either towards F, H, or towards E, G. Let them be produced and meet towards F, H at the point K.

Then, since EFK is in the plane AB, every point in EFK is in that plane; [XI. 1. therefore K is in the plane AB.

For the same reason K is in the plane CD. Therefore the planes AB, CD, being produced, meet one another. But they do not meet, since they are parallel by hypothesis. Therefore EF and GH, being produced, do not meet towards F, H. In the same manner it may be shewn that they do not meet towards E, G. But straight lines which are in the same plane, and which being produced ever so far both ways do not meet are parallel; therefore EF is parallel to GH.

Wherefore, if two parallel planes &c.