Elements of geometry and trigonometry . PROPOSITION IX. THEOREM. If two triangles have two sides of the one equal to two sides of the other, each to each, and the included angles unequal, the thi^d sides will be unequal ; and the greater side will belong to the triangle which has the greater included Let BAC and EDF be two triangles, havingthe side J^B^ DE, AC~DF, and the angleA>D;the\willBC>EF, ^* MakethefegleCAOB-D: ^akfN;AO-rDE,aT;d drawCdi The. BOOK I. lî) triangle GAC is equal to DEF, since, by construction, tlieyhave an equa angle in each, contained by equal sides, (
Elements of geometry and trigonometry . PROPOSITION IX. THEOREM. If two triangles have two sides of the one equal to two sides of the other, each to each, and the included angles unequal, the thi^d sides will be unequal ; and the greater side will belong to the triangle which has the greater included Let BAC and EDF be two triangles, havingthe side J^B^ DE, AC~DF, and the angleA>D;the\willBC>EF, ^* MakethefegleCAOB-D: ^akfN;AO-rDE,aT;d drawCdi The. BOOK I. lî) triangle GAC is equal to DEF, since, by construction, tlieyhave an equa angle in each, contained by equal sides, () ; therelbre CG is equal to EF. Now, there may be threecases in the proposition, according as the point G talis withoutihe triangle AIjC, or upon its base BC, or within it. First Case. The straight line GC<GI-f IC, and the straightline AB<AI + 1B; therefore, GC-hAB< GH AUIC + , which is the saiue thing, GC + AB<AG + BC. Take awayAB from the one side, and its equal AG from the other; andthere remains GCEF. ASecond Case. If the point Gfall on the side BC, it is evidentthat GC, or its equal EF, will beshorter than BC (Ax. 8.). Third Case. Lastly, if the point Gfall within the triangle BAC, we shallhave, by the preceding theorem, AG4-GC<AB4-BC; and, taking AG fromthe one, and its equal AB from the other,there will remain GC< BC orBOEF. B
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
