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.*

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.

Thus, in the triangle *ABK* the sum of the two angles *ABK* and *BAK* equals two right angles, which is impossible.

Therefore the planes *CD* and *EF* do not meet when produced. Therefore the planes *CD* and *EF* are parallel.

Therefore, *planes at right angles to the same straight line are parallel.*

Q.E.D.