Straight lines parallel to the same straight line are also parallel to one another.

Let each of the straight lines $AB$, $CD$ be parallel to $EF$;I say that $AB$ is also parallel to $CD$.For let the straight line $GK$ fall upon them.Then, since the straight line $K$ has fallen on the parallel straight lines $AB$, $EF$,the angle $AGK$ is equal to the angle $GHF$. [Prop. 1.29]Again, since the straight line $GK$ has fallen on the parallel straight lines $EF$, $CD$,the angle $GHF$ is equal to the angle $GKD$. [Prop. 1.29]But the angle $AGK$ was also proved equal to the angle $GKD$; [C.N. 1]and they are alternate.Therefore $AB$ is parallel to $CD$.Q.E.D.