If two straight lines are parallel and points are taken at random on each of them, then the straight line joining the points is in the same plane with the parallel straight lines.

Let *AB* and *CD* be two parallel straight lines, and let points *E* and *F* be taken at random on them respectively.

I say that the straight line joining the points *E* and *F* lies in the same plane with the parallel straight lines.

For suppose it is not, but, if possible, let it be in a more elevated plane as *EGF.* Draw a plane through *EGF.* Its intersection with the plane of reference is a straight line. Let it be *EF.*

Therefore the two straight lines *EGF* and *EF* enclose an area, which is impossible. Therefore the straight line joined from *E* to *F* is not in a plane more elevated. Therefore the straight line joined from *E* to *F* lies in the plane through the parallel straight lines *AB* and *CD.*

Therefore, *if two straight lines are parallel and points are taken at random on each of them, then the straight line joining the points is in the same plane with the parallel straight lines.*

Q.E.D.

Note that this proof assumes that every line lies in a plane, a conclusion that has not been justified.