| Planes to which the same straight line is at right angles are parallel. | ||
|
Let any straight line AB be at right angles to each of the planes CD and EF.
I say that the planes are parallel. | ||
| For, if not, then they meet when produced. Let them meet. Then they intersect as a straight line. Let it be GH. | XI.3 | |
| Take a point K at random on GH, and join AK and BK. | ||
| Now, since AB is at right angles to the plane EF, therefore AB is also at right angle to BK which is a straight line in the plane EF produced. Therefore the angle ABK is right. For the same reason the angle BAK is also right. | XI.Def.3 | |
| Thus, in the triangle ABK the sum of the two angles ABK and BAK equals two right angles, which is impossible. | I.17 | |
| Therefore the planes CD and EF do not meet when produced. Therefore the planes CD and EF are parallel. | XI.Def.8 | |
| Therefore, planes to which the same straight line is at right angles are parallel. | ||
| Q. E. D. | ||
Book XI Introduction - Proposition XI.13 - Proposition XI.15.