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. 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 Hint. — Bisect the vertical angle and prove that two symmetrical triangles 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, · bisectors,” perpendiculars," etc., are often used for arcs of great circles, arcs of great circles bisecting spherical angles, etc. 66 lines," " 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. 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. Ex. 1190. A central angle of a circle on a sphere is measured by the intercepted arc. Ex. 1191. If two small circles on a sphere intersect, their line of centres bisects the common chord at right angles. Ex. 1192. A radius perpendicular to a chord of a small circle on a sphere bisects the chord. Ex. 1193. In a circle on a sphere, equal chords are equidistant from the pole. 726. DEF. Two spherical polygons are vertical if their corresponding polyedral angles are vertical. B A В' Ex. 1194. Two vertical spherical triangles are symmetrical. POLAR TRIANGLES 727. DEF. If from the vertices of any spherical triangle as poles arcs of great circles are described, another triangle will be forined which is called the polar triangle of the first. Thus, if A is the pole of great circle B'C', B the pole of great circle A'C', and C the pole of great circle A'B', A A'B'C' . is the polar triangle of ABC. If from the poles A, B, and C entire circles should be described, eight triangles would be formed. The polar triangle is selected by the following method: Denote the vertex formed by the intersection of the arcs from B and C by A', then of the four points of intersection that one is A' whose distance from A is less than a quadrant. Similarly for B' and C'. PROPOSITION XIV. THEOREM 728. If one spherical triangle is the polar triangle of another, then the second spherical triangle is the polar triangle of the first. Hyp. A'B'C' is the polar triangle of ABC. To prove ABC is the polar triangle of A'B'C'. Proof. Since A is the pole of B'C', the distance AB' is a quadrant, and since C is a pole of A'B', the distance B'C is a quadrant. Therefore, B' is the pole of AC. (691) Similarly, A' is the pole of BC, and C' is the pole of AB. But the distances AA', BB' CC" are less than a quadrant. Hence ABC is the polar triangle of A'B'C'. Q.E.D. |