For let a point G be taken at random on EF, and from it let there be drawn GH, in the plane through EF, AB, at right angles to EF, and GK in the plane through FE, CD again at right angles to EF.
Now, since EF is at right angles to each of the straight lines GH, GK,
therefore EF is also at right angles to the plane through GH, GK. [XI. 4]
And EF is parallel to AB;
therefore AB is also at right angles to the plane through HG, GK. [XI. 8]
For the same reason CD is also at right angles to the plane through HG, GK;
therefore each of the straight lines AB, CD is at right angles to the plane through HG, GK.
But if two straight lines be at right angles to the same plane, the straight lines are parallel; [XI. 6]
therefore AB is parallel to CD.
Complete text
Book I