Planes Perpendicular to same Straight Line are Parallel

Proof

 * Euclid-XI-14.png

Let $AB$ be a straight line which is perpendicular to each of the planes $CD$ and $EF$.

It is to be demonstrated that $CD$ and $EF$ are parallel.

Suppose, to the contrary, that $CD$ and $EF$ are not parallel.

Then when produced they will meet.

From :
 * let the common section be the straight line $GH$.

Let $K$ be an arbitrary point on $GH$.

Let $AK$ and $BK$ be joined.

We have that $AB$ is perpendicular to $EF$.

So from :
 * $AB$ is perpendicular to the straight line $BK$ in the planes $EF$ produced.

Therefore $\angle ABK$ is a right angle.

For the same reason $\angle BAK$ is a right angle.

Thus, in $\triangle ABK$, there are two angles which are right angles.

From this is impossible.

Therefore $CD$ and $EF$ do not meet when produced.

So from :
 * $CD$ and $EF$ are parallel.