## Euclid, Elements of Geometry, Book I, Proposition 14 (Edited by Sir Thomas L. Heath, 1908)

### Proposition 14[Euclid, ed. Heath, 1908, on archive.org]

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.

Next: Proposition 15

Previous: Proposition 13

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