...SCH. Two polar triangles are also called supplemental triangles. Proposition 14. Theorem. 718. TJie sum of the angles of a spherical triangle is greater than two, and less than six, right angles. Hyp. Let A, B, C denote the three Z s of the spherical A ABC. To prove A + B + C > 180° and < 540°....

...may in fact be regarded as the polar triangle of A'BC. V.— Spherical Excess. We can now prove that the sum of the angles of a spherical triangle is greater than two right angles. For the sum of the angles of the triangle and the sides of the polar triangle is equal...

...triangle is one included by three arcs of great circles. It is a well-known geometrical theorem that the sum of the angles of a spherical triangle is greater than two right angles, and that the excess above two right angles bears the same ratio to a right angle as the...

...spherical polygon is at a quadrant's distance from every point in the side. PROPOSITION XI. 57O. Theorem : The sum of the angles of a spherical triangle is greater than two, and less than six right angles. Statement : Let ABC be any spherical triangle. The sum of its angles is less than six, and greater...

...two points of the arc of a great circle, it is the pole of that circle. PROPOSITION XI. 57O. Theorem: The sum of the angles of a spherical triangle is greater than two, and less than sijc right a ngles. Statement : Let ABC be any spherical triangle. The sum of its angles is less than...

...middle meridian. SPHEKICAL EXCESS or TRIANGLES ON THE EARTH'S SURFACE. 159. It is shown in geometry that the sum of the angles of a spherical triangle is greater than two right angles, and that the excess of this sum over two right angles is to eight right angles as the...

...Scholium. Polar triangles are sometimes called supplementary triangles. PROPOSITION XII. THEOREM. 280. The sum of the angles of a spherical triangle is greater than two, and less than six, right angles. Notation. Same as in § 278. To prove A + B+O 180° < 540°. Proof. A = 180° — >i', B = 180° —...

...of a convex spherical polygon is less than the circumference of a great circle. PROPOSITION XVIII. The sum of the angles of a spherical triangle is greater than two, and less than six, right angles. PROPOSITION XIX. Two symmetrical spherical triangles are equivalent. PROPOSITION XX. If two arcs of...

...Use the diagram of (209), and complete the demonstration by (45). PROPOSITION XXXIV. 211. THEOREM. The sum of the angles of a spherical triangle is greater than 180° and less than 540°. Now, A = 180° - B'C' . . . (205) and B = 180° - A'C' and C = 180° - A'B';...

...sides of a convex spherical polygon is less than the circumference of a great circle. PROPOSITION XXVI. The sum of the angles of a spherical triangle is greater than two and less than six right angles. PROPOSITION XXVII. Two symmetrical spherical triangles are equal in area. PROPOSITION XXVIII. PROPOSITION...