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Euclid's Elements: Book I: Proposition 14

Proposition 14: 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, C.N. 3 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.