(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, 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 exterior angle EGB
equal to the interior and opposite
angle GHD, or the
interior angles on the same side, namely
BGH, GHD,
equal to two right angles;

I say that AB is parallel
to CD.

For, since the angle EGB
is equal to the
angle GHD,

while the angle EGB
is equal to the angle AGH,
[I. 15]

the angle AGH is also equal to
the angle GHD;

and they are alternate;

therefore AB is parallel to
CD.
[I. 27]

Again, since the angles
BGH, GHD
are equal to two right angles, and the angles
AGH, BGH
are also equal to two right angles,
[I. 13]

the angles AGH,
BGH are equal to the angles
BGH, GHD.

Let the angle BGH be subtracted from
each;

therefore the remaining angle AGH
is equal to the remaining
angle GHD;

and they are alternate;

therefore AB is parallel to
CD.
[I. 27]

Therefore etc. Q.E.D.

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

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