PROPOSITION 30.¶

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,

[I. 29]the angle AGK is equal to the angle GHF.

Again, since the straight line GK has fallen on the parallel straight lines EF, CD,

[I. 29]the angle GHF is equal to the angle GKD.

But the angle AGK was also proved equal to the angle GHF;

[C. N. 1]therefore the angle AGK is also equal to the angle GKD;

and they are alternate.

Therefore AB is parallel to CD.[1]