Kitap 1
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Then, since AE is equal to EC, and BE to EF, the two sides AE, EB are equal to the two sides CE, EF respectively; and the angle AEB is equal to the angle FEC, for they are vertical angles. [I. 15] Therefore the base AB is equal to the base FC, and the triangle ABE is equal to the triangle CFE, and the remaining angles are equal to the remaining angles respectively, namely those which the equal sides subtend; [I. 4] therefore the angle BAE is equal to the angle ECF.
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But the angle ECD is greater than the angle ECF; [C. N. 5] therefore the angle ACD is greater than the angle BAE.
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Similarly also, if BC be bisected, the angle BCG, that is, the angle ACD [I. 15], can be proved greater than the angle ABC as well.
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Therefore etc.
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Q. E. D.
Proposition 17.
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Enunciation In any triangle two angles taken together in any manner are less than two right angles.
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Proof. Let ABC be a triangle; I say that two angles of the triangle ABC taken together in any manner are less than two right angles.
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For let BC be produced to D. [Post. 2]
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Then, since the angle ACD is an exterior angle of the triangle ABC,
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it is greater than the interior and opposite angle ABC. [I. 16] Let the angle ACB be added to each; therefore the angles ACD, ACB are greater than the angles ABC, BCA. But the angles ACD, ACB are equal to two right angles. [I. 13]
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Therefore the angles ABC, BCA are less than two right angles.
