Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/52

 greater than the base of the other, the angle contained by the sides of that which has the greater base, shall be greater than the angle contained by the sides equal to them, of the other. Let ABC, DEF be two triangles, which have the two sides AB, AC equal to the two sides DE, DF, each to each, namely, AB to DE, and AC to DF, but the base BC greater than the base EF: the angle BAC shall be greater than the angle EDF. For if not, the angle BAC must be either equal to the angle EDF or less than the angle EDF. But the angle BAC is not equal to the angle EDF, for then the base BC would be equal to the base EF; but it is not; therefore the angle BAC is not equal to the angle EDF. Neither is the angle BAC less than the angle EDF, for then the base BC would be less than the base EF; but it is not; therefore the angle BAC is not less than the angle EDF. And it has been shewn that the angle BAC is not equal to the angle EDF. Therefore the angle BAC is greater than the angle EDF. Wherefore, if two triangles &c. PROPOSITION 26. THEOREM. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, either the sides adjacent to the equal angles, or sides which are opposite to equal angles in each, then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other. Let ABC, DEF be two triangles, which have the angles ABC, BCA equal to the angles DEF, EFD, each