Straight lines which are parallel to the same straight line and are not in the same plane with it are also parallel to one another.
如果两条直线平行于同一条直线，且不与它共面，那么这两条直线也互相平行。
For let each of the straight lines AB, CD be parallel to EF, not being in the same plane with it; I say that AB is parallel to CD. 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.
设AB和CD都平行于EF，且不与EF共面。
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.
在EF上任取一点G，在过EF和AB的平面内作GH垂直于EF，在过FE和CD的平面内作GK垂直于EF。
[XI. 4] And EF is parallel to AB; therefore AB is also at right angles to the plane through HG, GK.
因为EF垂直于GH和GK，所以EF垂直于GH和GK所确定的平面。又因为EF平行于AB，所以AB也垂直于该平面。
[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.
同理，CD也垂直于该平面，因此AB和CD都垂直于同一平面，故AB平行于CD。