Heath ID: I.30

G.43

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. [G.42]

Again, since the straight line GK has fallen on the parallel straight lines EF, CD, the angle GHF is equal to the angle GKD. [G.42]

But the angle AGK was also proved equal to the angle GHF; therefore the angle AGK is also equal to the angle GKD; [G.1] and they are alternate.

Therefore AB is parallel to CD.

Q. E. D.

Note

20

The usual conclusion in general terms (Therefore etc.

) repeating the enunciation is, curiously enough, wanting at the end of this proposition.

Dependency Graph

Required for

G.58