Plane and solid geometry . al polygon is an arc of agreat circle (§ 936), the planes of these arcs meet at the centerof the sphere and form at that point a polyhedral angle, aspolyhedral Z 0-ABCD. This polyhedral angle and the spheri-cal polygon are very closely following are some of the moreimportant relations; the studentshould prove the correctness of each: 940. (a) 77ie sides of a sphericalpolygon have the same measures as thecorresponding face angles of the pfoly-hedral angle. (b) TJie angles of a sphericcd p^olygon have the same measuresas the corresponding dihedral angles of
Plane and solid geometry . al polygon is an arc of agreat circle (§ 936), the planes of these arcs meet at the centerof the sphere and form at that point a polyhedral angle, aspolyhedral Z 0-ABCD. This polyhedral angle and the spheri-cal polygon are very closely following are some of the moreimportant relations; the studentshould prove the correctness of each: 940. (a) 77ie sides of a sphericalpolygon have the same measures as thecorresponding face angles of the pfoly-hedral angle. (b) TJie angles of a sphericcd p^olygon have the same measuresas the corresponding dihedral angles of the poh/hedrcd angle. Thus, sides AB, BCy etc., of spherical polygon ABCD havethe same measures as face A AOB, BOC, etc., of polyhedralZ 0-ABCD] and spherical A ABC, BCD, etc., have the samemeasures as the dihedral A whose edges are OB, OC, etc. These relations make it ])ossible to establish certain prop-erties of spherical polygons from the corresponding knownpro])erties of the polyhedral angle, as in §§ 1)41 and BOOK IX 431 Proposition IX. Theorem 941. The sum of any two sides of a spherical triangleis greater than tJie third side.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
