Euclid,

ElementsI 18©

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

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Prop. 18: The larger side of any triangle subtends the larger angle.

(diagram 1) For let there be a triangle, ABG, having side AG larger than AB. (diagram 2) I say that an angle, that by ABG, is also larger than that by BGA. (diagram 3) For since AG is larger than AB, let an equal, AD, to AB be positioned, (diagram 4) and let BD be joined. (diagram 5) And since an angle, that by ADB, is exterior to a triangle, BGD, it is larger than the interior and opposite angle, that by DGB. (diagram 6) But the angle by ADB is equal to that by ABD, since AB is also equal to AD. (diagram 7) Therefore, the angle by ABD is also larger than that by AGB. Therefore, the angle by ABG is much larger than that by AGB. Therefore, the larger side of any triangle subtends the larger angle, just what it was required to show.

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