...equal sides are opposite equal angles. III. Any angle of a spherical triangle is less than 180°. IV. The sum of the angles of a spherical triangle is greater than 180° and less than 540° ; ie 180° < A + B + C< 540°. V. Any side of a spherical triangle is less...

...MN is the measure of the angle A, (page 68 ). Therefore a' + A = 180°, or A = 180° - a', etc. 71. The sum of the angles of a spherical triangle is greater than two and less than six right angles. Let ABC be a spherical triangle. To prove that 180° < A + B + С <...

...MN is the measure of the angle A, (page 68). Therefore a' + A = 180°, or A = 180° - a', etc. 71. The sum of the angles of a spherical triangle is greater than two and less than six right angles. Let ABC be a spherical triangle. To prove that 180° < A + B + С <...

...with its base 359°, and its other two sides each 90°, the sum of the sides being 539°. THEOREM. The sum of the angles of a spherical triangle is greater than 180° and less than 540°. It is for the purpose of proving this important fact that polar triangles...

...diagonal of a convex spherical polygon is as great as 180° of arc. PROPOSITION XXIX. 750. THEOREM. The sum of the angles of a spherical triangle is greater than 180° and less than 540°. Z-B+ZO 180° and LA + LB the polar of B = 180°— I', C = 540° Given A...

...9f,8 54, 2. 988. 54, 2. 985. 8. §309. 9. 10. § § 54, 54, 8 a. 3. 991. In § 949 it was proved that the sum of the angles of a spherical triangle is greater than 180° and less than 540°. Hence the spherical excess of a spherical triangle may vary from 0° to...

...B = 80°, prove that Z C > 10°. HINT. Construct the polar A A'B'C'. PROPOSITION XVII. THEOREM 754. The sum of the angles of a spherical triangle is greater than two and less than six right angles. Given ABC, a spherical triangle. To prove Z A + ZB + Z C> 180°, and...

...(a).] 2. The sum of the sides of a spherical polygon is less than 360°. [O12 (b). See also (a).] 3. The sum of the angles of a spherical triangle is greater than 180° and less than 540°. [K8.j 4. // A'B'C' is he polar triangle of ABC, then, reciprocally, ABC...

...plane parallel to the base is a figure of the same shape as the base (Prove for circular cone only) 7 The sum of the angles of a spherical triangle is greater than 180° but less than 540° 8 The planes which bisect the dihedral angles of a trihedral angle intersect...

...triangles can be changed into a new theorem by applying it first to the polar triangle ; thus from " The sum of the angles of a spherical triangle is greater than 180° " follows " The sum of the sides of a spherical triangle is less than 360°." (25) It may be...