Euclid,

ElementsI 29©

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

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Prop. 29: The straight-line falling into parallel straight-lines makes the alternate angles equal to one another and the external angle equal to the angle that’s internal and opposite and on the same sides and the angles that are interior and on the same sides equal to two right-angles.

(diagram 1) For let a straight-line, EZ, fall into parallel straight-lines AB, GD. I say that it makes the alternate angles, those by AHQ, HQD, equal to one another and the external angle, that by EHB, equal to the angle that’s internal and opposite and on the same sides, that by HQD, and the angles that are interior and on the same sides, those by BHQ, HQD, equal to two right-angles. (diagram 2) For if the angle by AHQ is unequal to that by HQD, one of them is larger. Let the angle by AHQ be larger. (diagram 3) Let a common, the angle by BHQ, be added. Therefore, the angles by AHQ, BHQ are larger than those by BHQ, HQD. But those by AHQ, BHQ are equal to two right-angles. Therefore, the angles by BHQ, HQD are smaller than two right-angles. (diagram 4) But straight-lines from angles smaller than two right-angles, on being extended to infinity, meet. Therefore, AB, G, on being extended to infinity will meet. But they do not meet because they are supposed parallel. Therefore, the angle by AHQ is not unequal to that by HQD. Therefore, it is equal. (diagram 5) But the angle by AHQ is equal to that by EHB. Therefore, the angle by EHB is also equal to that by HQD. (diagram 6) Let a common be added, that by BHQ. Therefore, the angles by EHB, BHQ are equal to those by BHQ, HQD. But EHB, BHQ are equal to two right-angles. Therefore, the angles by BHQ, HQD are also equal to two right-angles. Therefore, the straight-line falling into parallel straight-lines makes the alternate angles equal to one another and the external angle equal to the angle that’s internal and opposite and on the same sides and the angles that are interior and on the same sides equal to two right-angles, just what it was required to show.

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