(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

In any triangle two angles taken together in any manner are less than two right angles.

Let ABC be a triangle;

I say that two angles of the
triangle ABC taken
together in any manner are less than two right angles.

For let BC be produced to D.

Then, since the angle ACD is an
exterior angle of the
triangle ABC,

it is greater than the interior and opposite angle
ABC.
[I. 16]

Let the angle ACB be added
to each;

therefore the angles ACD,
ACB are greater than the angles
ABC, BCA.
But the angles ACD,
ACB are equal to two
right angles.
[I. 13]

Therefore the angles ABC,
BCA are less than two right angles.

Similarly we can prove that the angles BAC, ACB are also less than two right angles, and so are the angles CAB, ABC as well.

Therefore etc. Q.E.D.

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

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