If two planes which cut one another be at right angles to any plane, their common section will also be at right angles to the same plane.
如果两个相交平面都垂直于同一平面，那么它们的交线也垂直于该平面。
For let the two planes AB, BC be at right angles to the plane of reference, and let BD be their common section; I say that BD is at right angles to the plane of reference. For suppose it is not, and from the point D let DE be drawn in the plane AB at right angles to the straight line AD, and DF in the plane BC at right angles to CD.
设两平面AB和BC都垂直于参考平面，且BD是它们的交线。假设BD不垂直于参考平面。
Now, since the plane AB is at right angles to the plane of reference, and DE has been drawn in the plane AB at right angles to AD, their common section, therefore DE is at right angles to the plane of reference.
从点D在平面AB内作DE垂直于AD，在平面BC内作DF垂直于CD。
[XI. Def. 4] Similarly we can prove that DF is also at right angles to the plane of reference. Therefore from the same point D two straight lines have been set up at right angles to the plane of reference on the same side: which is impossible.
由于平面AB垂直于参考平面，且DE在平面AB内垂直于交线AD，故DE垂直于参考平面。同理，DF也垂直于参考平面。
[XI. 13] Therefore no straight line except the common section DB of the planes AB, BC can be set up from the point D at right angles to the plane of reference.
于是从同一点D在参考平面同侧有两条直线垂直于该平面，这与XI.13矛盾。因此只有交线BD垂直于参考平面。