Elements of geometry and trigonometry . For, lay the base ABCDE upon its equal abcdi ; these twobases will coincide. But the throe plane angles which form 148 GEOMETRY. the solid angle B, are respectively equal to the three planeangles, which form the solid angle b, namely, ABCzrra^c,ABGz=iabg, and GBC =gbc ; they are also similarly situated :hence the solid angles B and b are equal (Book VI. Prop. ) ; and therefore the side BG will fall on its equal bg. Itis likewise evident, that by reason of the equal parallelogramsABGF, abgf, the side GF will fall on its equal gfi and in thesame ma
Elements of geometry and trigonometry . For, lay the base ABCDE upon its equal abcdi ; these twobases will coincide. But the throe plane angles which form 148 GEOMETRY. the solid angle B, are respectively equal to the three planeangles, which form the solid angle b, namely, ABCzrra^c,ABGz=iabg, and GBC =gbc ; they are also similarly situated :hence the solid angles B and b are equal (Book VI. Prop. ) ; and therefore the side BG will fall on its equal bg. Itis likewise evident, that by reason of the equal parallelogramsABGF, abgf, the side GF will fall on its equal gfi and in thesame manner GH on gh ; hence, the plane of the upper base,FGHIK will coincide with the plane fghik (Book VI. Prop. II.). IC Yc. But the two upper bases being equal to their correspondinglower bases, are equal to each other : hence HI will coincidewith //./, IK with lA-, and KF with kf; and therefore the lateralfaces çf the prisms will coincide : therefore, the two prismscoinciding throughout are equal (Ax. 13.). Cor. Two right prisms, ivhich have equal bases and equal al-titudes, are equal. For, since the side AB is equal to ab, andthe altitude BG to bg, the rectangle ABGF will be equal toabgf; so also will the rectangle BGHC be equal to bghc ; andthus the three planes, which form the solid angle B, will beequal to the three which form the solid angle b. Hence thetwo prisms are equal. PROPOSITION VI. THEOREM. In every parallelopipedon the opposite planes are equal and parallel. By the definition of this solid, the basesABCD, EFGH, are equal parallelograms,and their sides arc parallel : it remainsonly to show, that the same is true of anytwo opposite lateral faces, such as AEHI),BFGC. Now AD is equal and paral
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
