Euclid,

ElementsI 17©

translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 17: The two angles of any triangle, taken in any way, are smaller than two right-angles.

(diagram 1) Let there be a triangle, ABG. I say that the two angles of triangle ABG, taken in any way, are less than two right angles. (diagram 2) For let BG be extended to D. And since an angle, that by AGD, is exterior to a triangle, ABG, it is larger than the interior and opposite angle, that by ABG. (diagram 3) Let a common be added, that by AGB. Therefore, the angles by AGD, AGB are larger than those by ABG, BGA. But the angles by AGD,AGB are equal to two ritght angles. Therefore, ABG, BGA are smaller than two right-angles. Similarly, in fact, we will show that the angles by BAG, AGB are also smaller than two right angles, and furthermore those by GAB, ABG. Therefore, the two angles of any triangle, taken in any way, are smaller than two right-angles, just what it was required to show.

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