If a straight line is at right angles to any plane, then all the planes through it are also at right angles to the same plane.

Let any straight line *AB* be at right angles to the plane of reference.

I say that all the planes through *AB* are also at right angles to the plane of reference.

Let the plane *DE* be drawn through *AB.* Let *CE* be the intersection of the plane *DE* and the plane of reference.

Take a point *F* at random on *CE,* and draw *FG* from *F* at right angles to *CE* in the plane *DE.*

Now, since *AB* is at right angles to the plane of reference, therefore *AB* is also at right angles to all the straight lines which meet it and lie in the plane of reference, so that it is also at right angles to *CE.* Therefore the angle *ABF* is right.

But the angle *GFB* is also right, therefore *AB* is parallel to *FG.*

But *AB* is at right angles to the plane of reference, therefore *FG* is also at right angles to the plane of reference.

Now a plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane. And *FG,* drawn in one of the planes *DE* at right angles to *CE,* the intersection of the planes, was proved to be at right angles to the plane of reference. Therefore the plane *DE* is at right angles to the plane of reference.

Similarly it can also be proved that all the planes through *AB* are at right angles to the plane of reference.

Therefore, *if a straight line is at right angles to any plane, then all the planes through it are also at right angles to the same plane.*

Q.E.D.