Planes to which the same straight line is at right angles will be parallel.
For let any straight line AB
be at right angles to each of the planes CD
; I say that the planes are parallel.
For, if not, they will meet when produced.
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
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
For the same reason the angle BAK
is also right.
Thus, in the triangle ABK
, the two angles ABK
are equal to two right angles: which is impossible. [I. 17
Therefore the planes CD
will not meet when produced; therefore the planes CD
are parallel. [XI. Def. 8
Therefore planes to which the same straight line is at right angles are parallel. Q. E. D.