Proposition I.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,

  • 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; I.c.n.1

and they are alternate.

Therefore AB is parallel to CD. 1

  • Q. E. D.

References

Footnotes

  1. Therefore...
    The usual conclusion in general terms (Therefore etc.) repeating the enunciation is, curiously enough, wanting at the end of this proposition. ↩