|Planes at right angles to the same straight line 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.Def.8|
|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 at right angles to the same straight line are parallel.|
|Q. E. D.|
Next proposition: XI.15