Euclid,

ElementsI 25©

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

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Prop. 25: If two triangles have the two sides respectively equal two sides and the base larger than the base, then they will have the angle that’s enclosed by the equal sides larger than the angle.

(diagram 1) Let there be two triangles, ABG, DEZ, having two sides, AB, AG, respectively equal to the two sides, DE, DZ, AB to DE and AG to DZ. But let a base, BG, be larger than a base, EZ. (diagram 2) I say that an angle, that by BAG, is larger than an angle, that by EDZ. For if not, either it is equal to it or it is smaller. (diagram 3) And so, the angle by BAG is not equal to that by EDZ. (diagram 4) For a base, BG, would also be equal to a base, EZ. But it is not. Therefore, an angle, that by BAG, is not equal to that by E DZ. (diagram 5) Truly, the angle by BAG is not smaller than that by EDZ either. (diagram 6) For a base, BG, would be smaller than a base, EZ. But it is not. Therefore, the angle by BAG is not smaller than that by EDZ. But it was shown that it is not equal either. (diagram 2) Therefore, the angle by BAG is larger than that by EDZ. Therefore, if two triangles have the two sides respectively equal two sides and the base larger than the base, then they will have the angle that’s enclosed by the equal sides larger than the angle, just what it was required to show.

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