Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.

Let *AB* and *CD* be equal and parallel, and let the straight lines *AC* and *BD* join them at their ends in the same directions.

I say that *AC* and *BD* are also equal and parallel.

Join *BC.*

Since *AB* is parallel to *CD,* and *BC* falls upon them, therefore the alternate angles *ABC* and *BCD* equal one another.

Since *AB* equals *CD,* and *BC* is common, the two sides *AB* and *BC* equal the two sides *DC* and *CB,* and the angle *ABC* equals the angle *BCD,* therefore the base *AC* equals the base *BD,* the triangle *ABC* equals the triangle *DCB,* and the remaining angles equals the remaining angles respectively, namely those opposite the equal sides. Therefore the angle *ACB* equals the angle *CBD.*

Since the straight line *BC* falling on the two straight lines *AC* and *BD* makes the alternate angles equal to one another, therefore *AC* is parallel to *BD.*

And it was also proved equal to it.

Therefore *straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.*

Q.E.D.

In general, given four points *A, B, C,* and *D,* exactly one of the three pairs of lines, *AB* and *CD, AC* and *BD,* and *AD* and *BC,* intersects. (If extended to infinite lines, all three pairs of lines might intersect, but as line segments only one pair does.) This statement belongs to the fundamental part of plane geometry that includes betweenness and sides of lines that wasn’t developed until the late nineteenth century.