Elements of geometry and trigonometry . 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 eq
Elements of geometry and trigonometry . 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^ PROPOSITION XI. THEOREM. Two tnangles on the same sphere, or on equal spheres, are equalin all their parts, when two angles and the included side of theone are equal to two angles and the included side of the other,each to each. For, one of these triangles, or the triangle symmetrical WMthit, may be placed on tlie other, as is done in the corres-ponding case of rectilineal triangles (Book I. Prop. VI.). PROPOSITION XII. THEOREM. If two tiiangles on the same sphere, or on equal spheres, have alltheir sides equal, each to each, their angles will likewise beequal, each to each, the equal angles lying opposite the equalsides. 190 GEOMETRY. This truth is evident from Prop. IX,where it was shown, that with three givensides AB, AC, BC, there can only be twotriangles ACB, ABD, differing as to theposition of their parts, and equal as to themagnitude of those parts. Hence thosetwo triangles, having all their sides re-spectively equal in both, must either beabsolutely equal, or at least symmetri
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