If two straight lines be parallel and points be taken at random on each of them, the straight line joining the points is in the same plane with the parallel straight lines.
若两条直线平行，且每条直线上各取任意一点，则连接这两点的线段与这两条平行线共面。
Let AB, CD be two parallel straight lines, and let points E, F be taken at random on them respectively; I say that the straight line joining the points E, F is in the same plane with the parallel straight lines.
设AB、CD为两条平行直线，其上分别任取点E、F。
For suppose it is not, but, if possible, let it be in a more elevated plane as EGF, and let a plane be drawn through EGF; it will then make, as section in the plane of reference, a straight line.
假设线段EF不在AB、CD所在平面内，而位于更高平面如EGF中。
[XI. 3] Let it make it, as EF; therefore the two straight lines EGF, EF will enclose an area: which is impossible.
过EGF作平面，与AB、CD所在平面交于一直线（依据XI.3），设该交线为EF。
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 is in the plane through the parallel straight lines AB, CD.
则EGF与EF两条直线围成一面积，此为不可能。故EF必在AB、CD所在平面内。