(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.

For let any straight line AB set up
on the straight line CD
make the angles CBA,
ABD;

I say that the angles CBA,
ABD are either two right angles or
equal to two right angles.

Now, if the angle CBA is equal to the angle ABD, they are two right angles. [Def. 10]

But, if not, let BE be drawn
from the point B at right angles to
CD;
[I. 11]

therefore the angles CBE,
EBD are two right angles.

Then, since the angle CBE is equal
to the two angles CBA,
ABE,

let the angle EBD be added
to each;

therefore the angles CBE,
EBD are equal to the three
angles CBA,
ABE, EBD.
[C.N. 2]

Again, since the angle DBA
is equal to the two angles
DBE,
EBA,

let the angle ABC be added
to each;

therefore the angles DBA,
ABC are equal to the three
angles DBE,
EBA, ABC.
[C.N. 2]

But the angles CBE,
EBD were also proved equal to the
same three angles;

and things which are equal to the same thing are also equal
to one another;
[C.N. 1]

therefore the angles CBE,
EBD are also equal to the
angles DBA,
ABC.

But the angles CBE,
EBD are two
right angles;

therefore the angles DBA,
ABC 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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