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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,

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.


Q. E. D.

1

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

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