Kitap 1
13
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.
13
Therefore etc.
13
Q. E. D.
Proposition 14.
14
Enunciation 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.
14
Proof. 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;
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I say that BD is in a straight line with CB.
14
For, if BD is not in a straight line with BC, let BE be in a straight line with CB.
14
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]
14
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.
14
Similarly we can prove that neither is any other straight line except BD. Therefore CB is in a straight line with BD.
14
Therefore etc.
14
Q. E. D.
Proposition 15.
15
Enunciation If two straight lines cut one another, they make the vertical angles equal to one another.
15
Proof. For let the straight lines AB, CD cut one another at the point E;
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I say that the angle AEC is equal to the angle DEB, and the angle CEB to the angle AED.
15
For, since the straight line AE stands on the straight line CD, making the angles CEA, AED, the angles CEA, AED are equal to two right angles [I. 13]
