Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/252

228 But ED is also at right angles to each of the two BD, CD;

therefore ED is at right angles to each of the three straight lines BD, AD, CD, at the point at which they meet;

therefore these three straight lines are all in the same plane. [XI. 5.

But AB is in the plane in which are BD, DA; [XI. 2.

therefore AB, BD, CD are in one plane.

And each of the angles ABD, CDB is a right angle;

therefore AB is parallel to CD. [I. 28.

Wherefore, if two straight lines &c.

PROPOSITION 7. THEOREM.

If two straight lines he parallel, the straight line drawn from any point in one to any point in the other, is in the same plane with the parallels.

Let AB, CD be parallel straight lines, and take any point E in one and any point F in the other: the straight line which joins E and F shall be in the same plane with the parallels.

For, if not, let it be, if possible, without the plane, as EGF; and in the plane ABCD, in which the parallels are, draw the straight EGF,EHF from E to F.

Then, since EGF is also a straight line, [Hypothesis.

the two straight lines EGF, EHF include a space between them; which is impossible. [Axiom 10.

Therefore the straight line joining the points E and F is not without the plane in which the parallels AB, CD are;

therefore it is in that plane.

Wherefore, if two straight lines &c.