Book 1 - Proposition 30
Straight lines parallel to the same straight line are also parallel to one another.

Let each of the straight lines AB and CD be parallel to EF.

I say that AB is also parallel to CD.

Let the straight line GK fall upon them. Since the straight line GK falls on the parallel straight lines AB and EF, therefore the angle AGK equals the angle GHF.

Again, since the straight line GK falls on the parallel straight lines EF and CD, therefore the angle GHF equals the angle GKD.

But the angle AGK was also proved equal to the angle GHF. Therefore the angle AGK also equals the angle GKD, and they are alternate.

Therefore AB is parallel to CD.

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