Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane

Proof

 * Euclid-XI-18.png

Let $AB$ be an arbitrary straight line which is perpendicular to the plane of reference.

It is to be demonstrated that every plane holding $AB$ is perpendicular to the plane of reference.

Let an arbitrary plane $DE$ be drawn through $AB$.

Let $CE$ be the common section of the plane $DE$ and the plane of reference.

Let $F$ be an arbitrary point on $CE$.

From :
 * let $FG$ be drawn in $DE$ perpendicular to $CE$.

We have that $AB$ is perpendicular to the plane of reference.

Therefore from :
 * $AB$ is perpendicular to all the straight lines which meet it and are in the plane of reference.

Therefore $AB$ is perpendicular to $CE$.

Therefore $\angle ABF$ is a right angle.

But $\angle GFB$ is also a right angle.

Therefore by :
 * $AB \parallel FG$

But $AB$ is perpendicular to the plane of reference.

Therefore from :
 * $FG$ is perpendicular to the plane of reference.

Thus the conditions are fulfilled for to apply:
 * $DE$ is perpendicular to the plane of reference.

As $DE$ is arbitrary, the result follows.