Elements of geometry and trigonometry . e, cd and de are all equal to eachother. Hence Ûie convex s(irf*ce of the frustum ABCDE-û?is formed by the equal trapcsoids E\a, &:c, and theperj>endicular distance between the parallel sides of either ofthese trapezoids is equal to Ff tlic slant height of the the area of either of the trapezoids, as AEcm, is equal to*(EA4-crf) X Ff (Book IV. Prop. VII.) : licnce die area of allof lliem, or the convex surface of the frustum, is equal to halfthe perimeters of the upper and lower bases multiplied by theslant heiffht PROPOSITION V. THEOREM. I
Elements of geometry and trigonometry . e, cd and de are all equal to eachother. Hence Ûie convex s(irf*ce of the frustum ABCDE-û?is formed by the equal trapcsoids E\a, &:c, and theperj>endicular distance between the parallel sides of either ofthese trapezoids is equal to Ff tlic slant height of the the area of either of the trapezoids, as AEcm, is equal to*(EA4-crf) X Ff (Book IV. Prop. VII.) : licnce die area of allof lliem, or the convex surface of the frustum, is equal to halfthe perimeters of the upper and lower bases multiplied by theslant heiffht PROPOSITION V. THEOREM. If the three planes which form a solid angle of a prism^ are equalto the three planes which form the solid angle of another prism^each to each^ and are like situated^ the two prisms will be equalto each oUicr, Let the base ABC DE be equal to the base aJycde^ the paral-lelogram AB(iF equal to the parallelogram ahgf and the par-allelogram BCHG equal to bchg-^ tlien will tlie prism ABCDE-Kbe equal to the prism 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
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
