(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.

For let the straight line EF
fall on the parallel straight lines
AB,
CD;

I say that it makes the alternate angles
AGH, GHD
equal, the exterior angle EGB
equal to the interior and opposite
angle GHD, and the interior angles on
the same side, namely BGH,
GHD, equal to two right angles.

For, if the angle AGH is unequal to the angle GHD, one of them is greater.

Let the angle AGH be greater.

Let the angle BGH be added
to each;

therefore the angles AGH,
BGH are greater than angles
BGH, GHD.

But the angles AGH,
BGH are equal to two right angles;
[I. 13]

therefore the angles
BGH, GHD
are less than two right angles.

But straight lines produced indefinitely from angles less than
two right angles meet
[Post. 5]
;
therefore
AB, CD,
if produced indefinitely, will meet;

but they do not meet, because they are by hypothesis parallel.

Therefore the angle AGH is not
unequal to the
angle GHD,

and is therefore equal to it.

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

therefore the angle EGB is also
equal to the
angle GHD.
[C.N. 1]

Let the angle BGH be added
to each;

therefore the angles
EGB, BGH
are equal to the angles
BGH, GHD.
[C.N. 1]

But the angles
EGB, BGH
are equal to two right angles
[I. 13]

therefore the angles BGH,
GHD are also equal to two right angles.

Therefore etc. Q.E.D.

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

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