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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.

1

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.

1 literally “let the angle EBD be added (so as to be) common,” κοινὴ προσκείσθω ἡ ὑπὸ ΕΒΔ. Similarly κοινὴ ἀφηρήσθω is used of subtracting a straight line or angle from each of two others. “Let the common angle EBD be added” is clearly an inaccurate translation, for the angle is not common before it is added, i.e. the κοινὴ is proleptic. “Let the common angle be subtracted” as a translation of κοινὴ ἀφηρήσθω would be less unsatisfactory, it is true, but, as it is desirable to use corresponding words when translating the two expressions, it seems hopeless to attempt to keep the word “common,” and I have therefore said “to each” and “from each” simply.

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

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