...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 +...

...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 360°, from which it follows...

...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 ABC. To Prove ZA ZC < 540°....

...75°, 85°, and 145°. Find the sides of its polar triangle. 434 l> PROPOSITION XIII. THEOREM 435 949. The sum of the angles of a spherical triangle is greater than 180° and less than 340°. Given spherical A ABC with sides denoted by a, b, and c. To prove ZA + Zn + ZC> 180° and <...

...90°, and 80°, respectively, find the sides of the polar triangle (in degrees). Why? Why? (c) § 361 367. Theorem IX. The sum of the angles of a spherical...prove that ZA + ZB+ZO 180° and < 540°. Proof. Let A A'B'C', with its sides denoted by a', b', and c', be the polar of A ABC. + b' = 180°, ZC + c' =...

...similar manner Z B + b' = 180°, and Z t7+ c' = 180°. The proof of (6) is left for the student. EXERCISE 367. Theorem IX. The sum of the angles of a spherical...spherical A ABC with the sides a, b, and c. To prove that Z^l + ZB+ZO 180° and < 540°. Proof. Let A A'B'C', with its sides denoted by a', b', and c', be the...

...triangle are equal, the triangle is isosceles. 454 SOLID CiEOMETRY PROPOSITION XXIX. 752. THEOREM. The sum of the angles of a spherical triangle is greater than 180° and less than 540°. Given A ABC. To Prove ZA+Z#+ZC> 180° and Z A + Z B + Z C < 540°. Proof. Sue. 1. Let A A'B'C' be the...

...(a).] 2. The sum of the sides of a spherical polygon is less than 360°. [Oi2 (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 is the polarofA'B'C. [K6.]...

...(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 is the polarofA'B'C. [K6.]...

...Ax. 9 Ax. 9 §654 and Similarly B + b' = 180°, and C + c' = 180°. PROPOSITION XIII. THEOREM 668. The sum of the angles of a spherical triangle is greater than 180° and less than 540°. Given a spherical triangle ABC, the letter at the vertex of each angle denoting its value in degrees,...