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" The sum of the angles of a spherical triangle is greater than two and less than six right angles ; that is, greater than 180 and less than 540. (gr). If A'B'C' is the polar triangle of ABC... "
An Elementary Treatise on Plane and Solid Geometry - Page 138
by Benjamin Peirce - 1871 - 150 pages
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Plane Geometry Developed by the Syllabus Method

Eugene Randolph Smith - Geometry, Plane - 1909 - 192 pages
...opposite sides are equal. (Use the polar triangles.) State this for a trihedral angle. 277. Theorem XVI. The sum of the angles of a spherical triangle is greater than one, and less than three, straight angles. Let the angles be A, B, C, the opposite sides be a, 6, c,...
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Plane and Spherical Trigonometry

Levi Leonard Conant - Trigonometry - 1909 - 316 pages
...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...
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The Elements of Plane and Spherical Trigonometry

John Gale Hun, Charles Ranald MacInnes - Trigonometry - 1911 - 236 pages
...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 <...
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The Elements of Plane and Spherical Trigonometry

John Gale HUN (and MAC INNES (Charles Ranald)), Charles Ranald MacInnes - Trigonometry - 1911 - 234 pages
...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 <...
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The Teaching of Geometry

David Eugene Smith - Geometry - 1911 - 339 pages
...concave, with its base 359, and its other two sides each 90, the sum of the sides being 539. '<f- THEOREM. The sum of the angles of a spherical triangle is greater than 180 and less than 540. It is for th'e purpose of proving this important fact that polar triangles...
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Plane and Solid Geometry: Suggestive Method

George Clinton Shutts - Geometry - 1912 - 392 pages
...side or 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. cGiven A ABC. To Prove ZA + Z-B+ZO 180 and ZA + LB + ZC < 540. Proof....
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Solid Geometry

Clara Avis Hart, Daniel D. Feldman, Virgil Snyder - Geometry, Solid - 1912 - 220 pages
...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...
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Schultze and Sevenoak's Plane and Solid Geometry

Arthur Schultze, Frank Louis Sevenoak - Geometry - 1913 - 488 pages
...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...
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Solid Geometry

George C. Shutts - 1913 - 212 pages
...spherical 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....
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Trigonometry

Alfred Monroe Kenyon, Louis Ingold - Trigonometry - 1913 - 132 pages
...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...
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