The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . PROPOSITIOX 26. THEOREM. If two triangles have two angles of the one equal to twoangles of the other, each to each, and one side equal toone side, namely, either the sides adjacent to the equalangles, or sides which are opx)osite to equal angles in each,then shall the other sides be equal, each to each, and alsothe third angle of the one equal to the third angle of theother. Let ABC, DEF be two triangles, which have theangles ABC
The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . PROPOSITIOX 26. THEOREM. If two triangles have two angles of the one equal to twoangles of the other, each to each, and one side equal toone side, namely, either the sides adjacent to the equalangles, or sides which are opx)osite to equal angles in each,then shall the other sides be equal, each to each, and alsothe third angle of the one equal to the third angle of theother. Let ABC, DEF be two triangles, which have theangles ABC, BCA equal to the angles DEF, EFD, each BOOK L 26. 29. \Construction.[ffypothesis. to each, namely, ABC to DBF, and BCA to BFD; and let them have also one side equal to one side ; and first letthose sides be equal which are adjacent to the equal anglesin the two triangles, namely, BC to BF: the other sidesshall be equal, each to each, namely, ^^ to BB, andyi C to i)i^, and the thirdangle BAG equal to the Athird angle BBF. For if AB be notequal to DB, one of themmust be gi-eater than theother. Let AB be thegreater, and make BGequal to BB, [I. 3. and join GC. Then because GB is equal to DB, BC to BF; the two sides GB, BC are equal to the two sides BB, BF,each to each ; and the angle GBC is equal to the angle BBF; {Hypothesis. therefore the triangle GBC is equal to the triangle DBF,and the other angles to the other angles, each to each, towhich the equal sides are opposite; [I. 4. therefore the angle GCB is equal to the angle BFE. But the angle BFE is equal to the angle A CB. [Hypothesis. Therefore the angle GCB is equal to the angle AC
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