Elements of geometry and trigonometry . ivalent to the right prism Bcd-h, But the two rightprisms Bad-h, Bcd-h, are equal, since they have the same alti-tude BF, and since their bases Bad, Bdc, are halves of thesame parallelogram (Prop. V. Cor.). Hence the two triaa- BOOK VII. 151 giilar prisms BAD-II, BDC-G, being e(iuivalcnt to tlie equalri<jht prisms, are equivalent to each otlier. Cor, Every triangular prism ABD-IIEF ishalf of the paral-lelopipedun AG described with the same solid angle A, andthe same edges AB, AD, AE. PROPOSITION VIII. THEOREM. If two parallelopipedons have a common ba
Elements of geometry and trigonometry . ivalent to the right prism Bcd-h, But the two rightprisms Bad-h, Bcd-h, are equal, since they have the same alti-tude BF, and since their bases Bad, Bdc, are halves of thesame parallelogram (Prop. V. Cor.). Hence the two triaa- BOOK VII. 151 giilar prisms BAD-II, BDC-G, being e(iuivalcnt to tlie equalri<jht prisms, are equivalent to each otlier. Cor, Every triangular prism ABD-IIEF ishalf of the paral-lelopipedun AG described with the same solid angle A, andthe same edges AB, AD, AE. PROPOSITION VIII. THEOREM. If two parallelopipedons have a common base, and their upperbases hi ike same phnic and between the same parallels, theywill be cquicalent. Let the f)aralleIopipe-dons AG, AL, have thecommon base AC, andtheir upper bases EG,MK, in the same plane,and between the sameparallels HL, EK; thenwill they be etjuivalent. There may be threecase^, according as El isgreater, less than, or equal to, EF ; but the demonstration isthe same for all. In the first place, then we shall show that. IS equal to the triangular the trianirular prism AEI-MDII,prism BFK-LCG. .Since AE is parallel to BF, and HE to GF, the angle. AEI---HIK, HEI = GIK, and IIEA^GIB. Also, since EF andIK are each equal to AB, they are ecjual to each oiher. Toeach add FI, and there will result El equal to FK : hence thetriangle AEI is ecjual to the triangle BFK (Bk. 1. Prop. V), andthe paralellograrn EM to the parallelogram FL. But the par-allelogram All is equal to the parallelogram CF (l*ro[). VI) :hence, the three planes which form the solid angle at E arerespectively ec|ual to the three which form the solid angle atF, and being like placed, the triangular prism AEl-M is equaltu the triangular prism BFK-L. But if the prism AEI-M is taken away from the solid AL,there will remain the parallelo[>iped<)n BADC-L ; and if thepnsm BFK-L is taken away from the same solid, tbere willremain the parallelopijKidon BADC-G ; hence those two paral-lelopipedons BADC-L, BAD
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