SPHERICAL POLYGONS 710. DEF. A spherical polygon is a portion of the surface of a sphere bounded by three or more arcs of great circles. The arcs are the sides, their points of intersection are the vertices, and the spherical angles formed by the sides are the angles of the polygon. 711. DEF. A diagonal of a spherical polygon is an arc joining any two non-adjacent vertices. B Thus, ABCD is a spherical polygon, AB, BC, etc., its sides, A, B, C, etc., its vertices, and ▲ ABC, BCD, etc., its angles. 712. DEF. A spherical triangle is a spherical polygon of three sides. It is called isosceles, equilateral, etc., in the same cases in which a plane triangle would be so called. The planes of the sides of a spherical polygon form at the center a polyedral angle (0- ABCD) which is said to correspond with the spherical polygon. The sides of the spherical polygon are measured by the sides of the corresponding polyedral angle, its angles are equal to the diedral angles of the corresponding polyedral angle. 713. REMARK. By means of the relations between the parts of a spherical polygon and those of its corresponding polyedral angle, we can deduce from any theorem of polyedral angles an analogous one of spherical polygons. 714. A spherical polygon is convex if its corresponding polyedral angle is convex. All spherical polygons are supposed to be convex polygons unless stated otherwise. 715. Spherical polygons are symmetrical if their corresponding polyedral angles are symmetrical. Evidently, their parts must be respectively equal, but follow in reverse order. In general, two symmetrical spherical polygons cannot be made to coincide. The sides of a spherical polygon are usually measured in degrees. PROPOSITION IX. THEOREM 716. The sum of two sides of a spherical triangle is greater than the third side. But the central angle is measured by the intercepted arc. PROPOSITION X. THEOREM 717. The sum of the sides of any spherical polygon is less than four right angles. Hyp. ABCDE is a spherical polygon. To prove AB + BC + CD + DE + EA < 360°. HINT. Construct the corresponding polyedral angle and compare Remark (713). PROPOSITION XI. THEOREM 718. Two triangles on the same sphere are equal: (1) If two angles and the included side of the one are respectively equal to two angles and the included side of the other, (2) If two sides and the included angle of the one are respectively equal to two sides and the included angle of the other, (3) If three sides of the one are respectively equal to three sides of the other, Provided the equal parts are arranged in the same order. HINT. - Prove the equality of the corresponding polyedral angles. 719. COR. Two symmetrical isosceles triangles are equal. PROPOSITION XII. THEOREM 720. Two triangles on the same sphere are symmetrical: (1) If two angles and the included side of the one are respectively equal to two angles and the included side of the other, (2) If two sides and the included angle of the one are respectively equal to two sides and the included angle of the other, (3) If three sides of the one are respectively equal to three sides of the other, Provided the equal parts are arranged in the reverse order. HINT.-Prove that the corresponding polyedral angles are symmetrical. 721. REMARK. — The equality of spherical angles and arcs is usually proven by means of equal or symmetrical triangles. PROPOSITION XIII. THEOREM 722. The base angles of an isosceles spherical triangle are equal. B D Bisect the vertical angle and prove that two symmetrical tri HINT. angles are formed. 723. COR. An equilateral spherical triangle is also equiangular. 724. REMARK. Many theorems of Spherical Geometry may be proved by methods analogous to those of Plane Geometry. 725. NOTE. In propositions relating to spherical figures, the words, "lines," bisectors," "perpendiculars," etc., are often used for arcs of 66 great circles, arcs of great circles bisecting spherical angles, etc. Ex. 1173. Every point in a perpendicular bisector of an arc of a great circle is equidistant from the ends of the arc. Ex. 1174. Two points equidistant from the ends of an arc of a great circle determine the perpendicular bisector of the arc. Ex. 1175. To bisect a spherical angle. Ex. 1176. To bisect an arc of a great circle. Ex. 1177. At a point in a given arc of a great circle, to draw a perpendicular to the arc. Ex. 1178. From a point without, to draw a perpendicular to a given great arc. Ex. 1179. If the opposite sides of a spherical quadrilateral are equal, the opposite angles are equal. Ex. 1180. Vertical spherical angles are equal. Ex. 1181. If two semicircumferences have common ends, they include equal angles. Ex. 1182. If the opposite sides of a spherical quadrilateral are equal, the diagonals bisect each other. Ex. 1183. To circumscribe a circle about a spherical triangle. Ex. 1184. At a given point in a great circle, to draw an angle equal to a given angle. Ex. 1185. To construct a spherical triangle having given two sides and the included angle. Ex. 1186. To construct a spherical triangle having given three sides. Ex. 1187. To construct a spherical triangle having given the base, the altitude, and the median corresponding with the base. Ex. 1188. The three bisectors of a spherical triangle meet in a point. Ex. 1189. The bisectors of the base angles of an isosceles spherical triangle are equal. |