## Book I, Proposition 30

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

| 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 GK has fallen on the parallel straight lines AB, EF, the angle AGK is equal to the angle GHF. [I. 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. [I. 29] But the angle AGK was also proved equal to the angle GHF; therefore the angle AGK is also equal to the angle GKD; [C.N. 1] and they are alternate. Therefore AB is parallel to CD. |