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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 and Solid Geometry

William Betz - Geometry - 1916 - 536 pages
...there congruence theorems for triangles in plane geometry corresponding to all of these cases? 844. The sum of the angles of a spherical triangle is greater than two and less than six right angles. Given the spherical triangle ABC, in which A, B, and C respectively...
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Solid Geometry

John Charles Stone, James Franklin Millis - Geometry, Solid - 1916 - 196 pages
...their polar triangles are mutually equilateral ; and conversely. The proof is left to the student. 474. Theorem. — The sum of the angles of a spherical triangle is greater than 180 and less than 540. Hypothesis. A ABC is any spherical triangle. Conclusion. ZA + ZB + Z C>...
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Solid Geometry

William Betz, Harrison Emmett Webb - Geometry, Solid - 1916 - 214 pages
...are respectively equal, and they are either congruent or symmetric. PROPOSITION XVIII. THEOREM 844. The sum of the angles of a spherical triangle is greater than two and less than six right angles. Given the spherical triangle ABC, in which A, J5, and C respectively...
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Plane and Spherical Trigonometry

Claude Irwin Palmer, Charles Wilbur Leigh - Logarithms - 1916 - 188 pages
...E-OD-C, and CAB and C-OE-D. 110. The sum of the sides of a spherical triangle is less than 360. The sum of the angles of a spherical triangle is greater than 180 and less than 540. It is evident that the sides and angles of a spherical triangle can be greater...
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Solid Geometry

John H. Williams, Kenneth P. Williams - Geometry, Solid - 1916 - 184 pages
...are 84, 100, and 110, what are the angles of the polar triangle? PROPOSITION XIV. THEOREM 750. The sum of the angles of a spherical triangle is greater than 180 and less than 540. a' Let ABC be a spherical triangle and A'B'C' its polar triangle. To prove...
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Geodetic Surveying

Edward Richard Cary - Geodesy - 1916 - 302 pages
...where e is the difference above stated. o|+6|-+c|•+dJ=360'> FIG. 47. 57. Spherical Excess. The sum of the angles of a spherical triangle is greater than 180. The excess becomes appreciable when the sides are from 4 to 5 miles long. The equation* for...
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Solid Geometry

Fletcher Durell, Elmer Ellsworth Arnold - Geometry, Solid - 1917 - 220 pages
...circumference of a great circle of the sphere. PROPOSITION XVI. THEOREM 1. Post. 1. 2. Why? 3. 677. QED 691. The sum of the angles of a spherical triangle is greater than two, and less than six, right angles. Given the spherical triangle ABC. To prove A + B + C > 180 and <...
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Plane and Solid Geometry

Claude Irwin Palmer, Daniel Pomeroy Taylor - Geometry - 1918 - 436 pages
...is opposite the greater angle; and conversely. Proof of theorem is similar to that of 184. 858. Theorem. The sum of the angles of a spherical triangle is greater than 180 and less than 540. Given the spherical AABC with the letter at each vertex of an angle denoting...
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Schultze and Sevenoak's Plane and Solid Geometry

Arthur Schultze, Frank Louis Sevenoak - Geometry - 1918 - 486 pages
...= 80, prove that Z C> 10. II INT. 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. A' K Given ABC, a spherical triangle. To prove ZA + ZB + Z C> 180,...
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Solid Geometry

Claude Irwin Palmer - Geometry, Solid - 1918 - 192 pages
...equal, the triangle is isosceles. 3. If a spherical triangle is equiangular it is equilateral. 858. Theorem. The sum of the angles of a spherical triangle is greater than 180 and less than 540. Given the spherical AABC with the letter at each vertex of an angle denoting...
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