Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/51

 greater than the base EF. Of the two sides DE, DF, let DE be the side which is not greater than the other. At the point D in the straight line DE, make the angle EDG equal to the angle BAC, and make DG equal to AC or DF, and join EG, GF. Because AB is equal to DE, and AC to DG; the two sides BA, AC are equal to the two sides ED, DG, each to each; and the angle BAC is equal to the angle EDG; therefore the base BC is equal to the base EG. And because DG is equal to DF, the angle DGF is equal to the angle DFG. But the angle DGF is greater than the angle EGF. Therefore the angle DFG is greater than the angle EGF. Much more then is the angle EFG greater than the angle EGF. And because the angle EFG of the triangle EFG is greater than its angle EGF, and that the greater angle is subtended by the greater side, therefore the side EG is greater than the side EF. But EG was shewn to be equal to BC; therefore BC is greater than EF. Wherefore, if two triangles &c. PROPOSITION 25. THEOREM. If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one