If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.

For let the straight line $EF$ falling on the two straight line $AB$, $CD$ make the alternate angles $AEF$, $EFD$ equal to one another;I say that $AB$ is parallel to $CD$.For, if not, $AB$, $CD$ when produced will will meet either in the direction of $B$, $D$ or towards $A$, $C$.Let them be produced and meet, in the direction of $B$, $D$, at $G$.Then, in the triangle $GEF$,the exterior angle $AEF$ is equal to the interior and opposite angle $EFG$ :which is impossible. [Prop. 1.16]Therefore $AB$, $CD$ when produced will not meet in the direction of $B$, $D$.Similarly it can be proved that neither will they meet towards $A$, $C$.But straight lines which do not meet in either direction are parallel; [Def. 23]therefore $AB$ is parallel to $CD$.Therefore etc.Q.E.D.