Kitap 1
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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.
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Similarly we can prove that the angles BAC, ACB are also less than two right angles, and so are the angles CAB, ABC as well.
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Therefore etc.
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Q. E. D.
Proposition 18.
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Enunciation In any triangle the greater side subtends the greater angle.
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Proof. For let ABC be a triangle having the side AC greater than AB;
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I say that the angle ABC is also greater than the angle BCA.
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For, since AC is greater than AB, let AD be made equal to AB [I. 3], and let BD bejoined.
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Then, since the angle ADB is an exterior angle of the triangle BCD,
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it is greater than the interior and opposite angle DCB. [I. 16]
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But the angle ADB is equal to the angle ABD, since the side AB is equal to AD; therefore the angle ABD is also greater than the angle ACB; therefore the angle ABC is much greater than the angle ACB.
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Therefore etc.
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Q. E. D.
Proposition 19.
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Enunciation In any triangle the greater angle is subtended by the greater side.
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Proof. Let ABC be a triangle having the angle ABC greater than the angle BCA;
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I say that the side AC is also greater than the side AB.
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For, if not, AC is either equal to AB or less.
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Now AC is not equal to AB; for then the angle ABC would also have been equal to the angle ACB; [I. 5] but it is not; therefore AC is not equal to AB.
