Page:Euclid's Elements 1714 Barrow translation.djvu/33

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EUCLIDE's Elements.

line CF cutting the ſides of the angle given any ways; b make AG—CD;upon AG c raiſe a trian- b 3. I gle equilateral to the former CDF, ſo that AH be c 21. I. equal to DF, and GH to CF. then ſhall you have the angle A d = D. Which was to he done.d 8. 1.

PROP. XXIV.

If two triangles ABC, DEF have two ſides of the one triangle AB, AC equal to two ſides of the other triangle DE, DF, each to other, and have the angle A greater than the angle EDF contained under the equal right lines, they ſhall alſo have the baſe BC greater than the baſe EF.

a Let the angle EDG be made equal to A,a 23. 1. and the fide DG b = DF c = AC; and let EG,b 1. and FG be joined. c hyp. 1. Caſe. Iſ EG fall above EF ; becauſe AE d= d hyp. DE, and ACe = DG, and the angle Ae = EDG, e conjtr. f therefore is BC = EG. But becauſe DFe = DG, f 4. 1. g, therefore is the angle DFG = DGF; h therefore g 5. 1. is the angle DFG = FFG, and by conſequence h 9. ax. the angle EFGb = EGF, k wherefore EG(BC) k 19. 1. = EF.

2.Caſe.

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