(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another.

For with any straight line AB,
and at the point B
on it, let the two straight lines
BC, BD
not lying on the same side make the adjacent angles
ABC, ABD
equal to two right angles;

I say that BD is in a
straight line with CB.

For, if BD is not in a straight line with BC let BE be in a straight line with CB.

Then, since the straight line AB
stands on the straight line
CBE,

the angles ABC,
ABE are equal to two right angles;
[I. 13]

But the angles ABC,
ABD are also equal to two
right angles;

therefore the angles CBA,
ABE are equal to the angles
CBA,
ABD.
[Post. 4
and
C.N. 1]

Let the angle CBA be subtracted
from each;

therefore the remaining angle ABE
is equal to the remaining angle
ABD,

the less to the greater: which is impossible.

Therefore BE is not in a straight
line with CB.

Similarly we can prove that neither is any other straight line
except BD.

Therefore CB is in a straight line
with BD.

Therefore etc. Q.E.D.

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

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