(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.

For let the straight line EF
falling on the two straight lines AB,
CD make the alternate angles
AEF, EFD
equal to one another;

I say that AB is parallel to
CD.

For, if not, AB, CD when produced will meet either in the direction of B, D or towards A, C.

Let them be produced and meet, in the direction of B, D, at G.

Then, in the triangle GEF,
the exterior angle AEF is equal
to the interior and opposite
angle EFG:

which is impossible.
[I. 16]
Therefore
AB, CD
when produced will not meet in the direction
of B, D.

Similarly it can be proved that neither will they meet towards A, C.

But straight lines which do not meet in either direction are
parallel;
[Def. 23]

therefore AB is
parallel to CD.

Therefore etc. Q.E.D.

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

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