Elements of geometry and trigonometry . maybeconceived as formed by the three plane angles AOB, AOD,BOD. And because the sides of the triangle ABC are equalto those of the triangle ADB, the plane angles forming the oneof these solid angles, must be equal to the plane angles formingthe other, each to each. But in that case we have shown thatthe planes, in which the equal angles lie, are equally inclinedto each other (Book YI. Prop. XXI.) ; hence all the angles ofthe spherical triangle DAB are respectively equal to those ofthe triangle CAB, namely, DAB=BAC, DBA=ABC, andADB = ACB; hence the sides
Elements of geometry and trigonometry . maybeconceived as formed by the three plane angles AOB, AOD,BOD. And because the sides of the triangle ABC are equalto those of the triangle ADB, the plane angles forming the oneof these solid angles, must be equal to the plane angles formingthe other, each to each. But in that case we have shown thatthe planes, in which the equal angles lie, are equally inclinedto each other (Book YI. Prop. XXI.) ; hence all the angles ofthe spherical triangle DAB are respectively equal to those ofthe triangle CAB, namely, DAB=BAC, DBA=ABC, andADB = ACB; hence the sides and the angles of the triangleADB are equal to the sides and the angles of the triangle ACB. Scholium. The equality of these triangles is not, however,an absolute equality, or one of superposition ; for it would beimpossible to apply them to each other exactly, unless theywere isosceles. The equality meant here is what we havealready named an equality by symmetry ; therefore we shallcall the triangles ACB, ADB, symmetrical EOOK IX. 19â PROPOSITION X. THEOREM. Two triangles on the same sphere, or on equal spheres, are equalin all their ^arts, when two sides and the included angle of theone are equal to two sides and the included angle of the other,each to each. Suppose the side AB = EF, the sideAC = EG, and the angle BAC=PEG ;then will the two triangles be equalin all their pait;^. For, the triangle EFG may beplaced on the triangle ABC, or onABD symmetrical with ABC, just astwo rectilineal triangles are placedupon each other, when they have anequal angle included between equal sides. Hence all the partsof the triangle EFG will be equal to all the parts of the trian-gle ABC ; that is, besides the three parts equal by hypothesis,we shall have the side BC = FG, the angle ABC=EFG, andthe angle ACB^EGF.
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
