Elements of geometry and trigonometry . equal and parallelto AB. For a similar rea-son, EF is equal and par-allel to AB ; hence also CDis equal and parallel toEF ; hence the figureCEFD is a parallelogram,and the side CE is equaland paiallel to DF; therefore the triangles CAE, DBF, havetheir corresponding sides equal; therefore the anï^le CAE —DBF. *^ - i Again, the plane ACE is parallel to the plane BDF. Forsuppose (he plane drawn throuirh the point A, parallel to BDF,were to meet the lines CD, EF, in points difierent from C andE, for instance in G and II ; then, the three lines AB, GD, FH,wou
Elements of geometry and trigonometry . equal and parallelto AB. For a similar rea-son, EF is equal and par-allel to AB ; hence also CDis equal and parallel toEF ; hence the figureCEFD is a parallelogram,and the side CE is equaland paiallel to DF; therefore the triangles CAE, DBF, havetheir corresponding sides equal; therefore the anï^le CAE —DBF. *^ - i Again, the plane ACE is parallel to the plane BDF. Forsuppose (he plane drawn throuirh the point A, parallel to BDF,were to meet the lines CD, EF, in points difierent from C andE, for instance in G and II ; then, the three lines AB, GD, FH,would he equal (Prop. XII.): but the lines AB, CD, EF, arealready known to be equal; hence CD = GD, and FII = EF,which is absurd ; hence the plane ACE is parallel to BDF. Cor. If two [)arallfl pianos MX, PQ are met bv two otherplanes CAIU), EABF, the aii<;lcs CAM DBF, formed by theintersections of the parallel i)laneswill be equal ; for. the inter-section AC is parallel to BD, and AE to BF (Prop. X.) ; there-fore the angle CAE^ PROPOSITION XIV. THEOREM. If three straight lines, not situatrd in the same plane, are eqnal(nul parallel, the opposite triauL^les formed hy join inî: the er-tremitirs of these lines will be equal, and their planes will Iteparallel. ^36 GEOMETRY. Let AB, CD, EF, be thelines. Since AB is equal andparallel to CD, the figureABDC is a parallelogram ;hence the side AC is equaland parallel to BD. For alike reason the sides AE,BF, are equal and parallel,as also CE, DF ; thereforethe two triangles ACE, BDF,are equa^ ; hence, by the lastProposition, their planes areparallel.
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