Kitap 1
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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.
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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.
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For, if BD is not in a straight line with BC, let BE be in a straight line with CB.
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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]
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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.
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Similarly we can prove that neither is any other straight line except BD. Therefore CB is in a straight line with BD.
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Therefore etc.
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Q. E. D.
Proposition 15.
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Enunciation If two straight lines cut one another, they make the vertical angles equal to one another.
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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.
