Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/259

Rh PROPOSITION 14. THEOREM.

Planes to which the same straight line is perpendicular are parallel to one another.

Let the straight line AB he perpendicular to each of the planes CD and EF: these planes shall be parallel to one another.

For, if not, they will meet one Smother when produced; let them meet, then their common section will be a straight line;

let GH be this straight line; in it take any point K, and join AK, BK.

Then, because AB is perpendicular to the plane EF, [Hyp. it is perpendicular to the straight line BK which is in that plane; [XI. Definition 8.

therefore the angle ABK is a right angle. For the same reason the angle BAK is a right angle. Therefore the two angles ABK, BAK of the triangle ABK are equal to two right angles; which is impossible. [I. 17 Therefore the planes CD and EF, though produced do not meet one another; that is, they are parallel. [XI. Definition 8.

Wherefore, planes &c. q.e.d.

PROPOSITION 15. THEOREM.

If two straight lines which meet one another, be parallel to two other straight lines which meet one another, but are not in the same plane with the first two, the plane pass-ing through these is parallel to the plane passing through the others.