Elements of geometry and trigonometry . n base of the two par-allelopipedons AG, AL ;since they have the samealtitude, their upper basesEFGH,IKLM,willbeinthe same plane. Also thesides EF and AB will beequal and parallel, as wellas IK and AB ; hence EFis equal and parallel toIK; for a like reason, GFis equal and parallel toLK. Let the sides EF, GH, be produced, and likewise KL,IM, till by their intersections they form the parallelogramNOPQ ; this parallelogram will evidently be equal to eitherof the bases EFGH, IKLM. Now if a third parallelopipedonbe conceived, having for its lower base the par
Elements of geometry and trigonometry . n base of the two par-allelopipedons AG, AL ;since they have the samealtitude, their upper basesEFGH,IKLM,willbeinthe same plane. Also thesides EF and AB will beequal and parallel, as wellas IK and AB ; hence EFis equal and parallel toIK; for a like reason, GFis equal and parallel toLK. Let the sides EF, GH, be produced, and likewise KL,IM, till by their intersections they form the parallelogramNOPQ ; this parallelogram will evidently be equal to eitherof the bases EFGH, IKLM. Now if a third parallelopipedonbe conceived, having for its lower base the parallelogramABCD, and NOPQ for its upper, the third parallelopipedonwill be equivalent to the parallelopipedon AG, since with thesame lower base, their upper bases lie in the same planeand between the same parallels, GQ, FN (Prop. VIIL).For the same reason, this third parallelopipedon will also beequivalent to the parallelopipedon AL ; hence the two paral-lelopipedons AG, AL, which have the same base and thesame altitude, are PROPOSITION X. THEOREM. Any parallelopipedon may he changed into an equivalent rectan-gular parallelopipedon having the same altitude and anequivalent hase. BOOK VII. 153
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