Elements of geometry and trigonometry . BOOK VI. 130 mains AD<AC. The two sides AS, SD, are equal lo thetwo AS, SC ; the third side AD is less than the third side AC ;Ihereture the an;rle ASD<ASC (Book I. Prop. IX. Sch.).Adding BSl) = BSC, we shall have ASD + BSD or ASB<ASC + B« PROPOSITION XX. THEOREM. • Tlie sum of the plane angles which form a solid angle is alwaysless than four right angles. Cut the solid angle S by any planeABCDE ; from O, a point in that plane,draw to the several angles the straightlines AG, OB, OC, 01), OE. The sum of the angles of the trianglesASB, BSC, tkc
Elements of geometry and trigonometry . BOOK VI. 130 mains AD<AC. The two sides AS, SD, are equal lo thetwo AS, SC ; the third side AD is less than the third side AC ;Ihereture the an;rle ASD<ASC (Book I. Prop. IX. Sch.).Adding BSl) = BSC, we shall have ASD + BSD or ASB<ASC + B« PROPOSITION XX. THEOREM. • Tlie sum of the plane angles which form a solid angle is alwaysless than four right angles. Cut the solid angle S by any planeABCDE ; from O, a point in that plane,draw to the several angles the straightlines AG, OB, OC, 01), OE. The sum of the angles of the trianglesASB, BSC, tkc. formed about the vertexS, is equal to the sum of the angles of anequal number of triangles AOB, BOC, & about the point O. But at thepoint B the sum of the angles ABO, OBC,equal to ABC, is less than the sum of theangles ABS, SBC (Prop. XIX.) ; in the same manner at thepoint C we have BCO-rOCD< BCS-fSCl); and so with aiithe angles of the polygon ABCDE: whence it follows, that thesum of all the angles at the bases of the triangles whose vertexis in O, is less than the sum of the angles at the bases of thetriangles whose vertex is in S ; hence to make up the dt-li-cicncy, the sum of the angles formed about the point O, isgreater than the sum of the angles fjrmcd al)ou
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
