If two parallel planes be cut by any plane, their common sections are parallel.
如果两个平行平面被任意平面所截，那么它们的交线互相平行。
For let the two parallel planes AB, CD be cut by the plane EFGH, and let EF, GH be their common sections; I say that EF is parallel to GH. For, if not, EF, GH will, when produced, meet either in the direction of F, H or of E, G. Let them be produced, as in the direction of F, H, and let them, first, meet at K.
设两平行平面AB、CD被平面EFGH所截，交线分别为EF、GH。
Now, since EFK is in the plane AB, therefore all the points on EFK are also in the plane AB. [XI. 1] But K is one of the points on the straight line EFK; therefore K is in the plane AB.
假设EF与GH不平行，则延长后将在F、H方向或E、G方向相交。
For the same reason K is also in the plane CD; therefore the planes AB, CD will meet when produced. But they do not meet, because they are, by hypothesis, parallel; therefore the straight lines EF, GH will not meet when produced in the direction of F, H. Similarly we can prove that neither will the straight lines EF, GH meet when produced in the direction of E, G.
设它们在F、H方向延长交于K。由于EFK在平面AB内，K也在平面AB内；同理K也在平面CD内，故平面AB、CD延长后相交，与假设平行矛盾。
But straight lines which do not meet in either direction are parallel. [I. Def. 23] Therefore EF is parallel to GH.
同理可证在E、G方向也不相交。因此EF与GH平行。