Planes to which the same straight line is at right angles will be parallel.
如果两个平面垂直于同一条直线，那么这两个平面互相平行。
For let any straight line AB be at right angles to each of the planes CD, EF; I say that the planes are parallel. For, if not, they will meet when produced.
设直线AB垂直于平面CD和EF，假设平面CD和EF不平行，则它们延长后相交。
Let them meet; they will then make, as common section, a straight line. [XI. 3] Let them make GH; let a point K be taken at random on GH, and let AK, BK be joined.
设交线为GH，在GH上任取一点K，连接AK和BK。
Now, since AB is at right angles to the plane EF, therefore AB is also at right angles to BK which is a straight line in the plane EF produced; [XI. Def. 3] therefore the angle ABK is right. For the same reason the angle BAK is also right.
因为AB垂直于平面EF，所以AB垂直于平面EF内的直线BK，角ABK为直角；同理，角BAK也为直角。
Thus, in the triangle ABK, the two angles ABK, BAK are equal to two right angles: which is impossible. [I. 17] Therefore the planes CD, EF will not meet when produced; therefore the planes CD, EF are parallel.
因此三角形ABK中有两个直角，与三角形内角和定理矛盾，故假设不成立，平面CD和EF平行。