(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.

Let ABC be a triangle,
and let one side of it BC be
produced to D;

I say that the exterior angle ACD
is equal to the two interior and opposite angles
CAB, ABC,
and the three interior angles of the triangle
ABC, BCA,
CAB are equal to two right angles.

For let CE be drawn through the point C parallel to the straight line AB. [I. 31]

Then since AB is parallel
to CE,

and AC has fallen
upon them,

the alternate angles BAC,
ACE are
equal to one another.
[I. 29]

Again, since AB is
parallel to CE,

and the straight line BD has fallen
upon them,

the exterior angle ECD is equal to
the interior and opposite
angle ABC.
[I. 29]

But the angle ACE was also proved
equal to the angle BAC;

therefore the whole angle ACD
is equal to the two interior and opposite angles
BAC, ABC.

Let the angle ACB be added
to each;

therefore the angles ACD,
ACB are equal to the three
angles ABC,
BCA, CAB.

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