Elements of geometry and trigonometry . he point P, erect a per-pendicular to the plane MN ; thatperpendicular must be at once inthe plane AB and in tlie plane Al>(Prop. XVII. Cor.) ; therefore itis theii common intersection AP. M w D B- y E ijNT PROPOSITION XIX. THEOREM. If a solid angle is formed by three plane angles, the sum of anytwo of these angles will be greater than the third. The proposition requires demonstra-tion only when the plane angle, whichis compared to the sum of the otliertwo, is greater than either of suppose the solid angle S tobe formed by three plane a
Elements of geometry and trigonometry . he point P, erect a per-pendicular to the plane MN ; thatperpendicular must be at once inthe plane AB and in tlie plane Al>(Prop. XVII. Cor.) ; therefore itis theii common intersection AP. M w D B- y E ijNT PROPOSITION XIX. THEOREM. If a solid angle is formed by three plane angles, the sum of anytwo of these angles will be greater than the third. The proposition requires demonstra-tion only when the plane angle, whichis compared to the sum of the otliertwo, is greater than either of suppose the solid angle S tobe formed by three plane angles ASB,ASC, BSC, whereof the angle ASB isthe greatest ; we are to show thatASB<ASC + BSC. In the plane ASB make the angle BSD = BSC, draw thestraight line ADB at pleasure; and having taken SC = SD,draw AC, BC. The two sides BS, SD, are equal to the two BS, SC ; theangle BSD=BSC ; therefore the triangles BSD, BSC, areequal; therefore BD=BC. ButABfrom the one side, and from the other its equal BC, there re. 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«C.
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
