(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

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.

Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)

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