THE RELATION OF SPHERICAL POLYGONS TO POLYHEDRAL ANGLES. If the center of a sphere is at the vertex of a pyramidal space, the pyramidal surface cuts from the spherical surface two spherical polygons. But the equal elements of these polygons are arranged in reverse order. And as the polyhedral angles are called opposite and are proved (VI, th. 25) symmetric, so the spherical polygons are called opposite spherical polygons. And since these have just been shown to have their corresponding elements equal but arranged in reverse order, they are called symmetric spherical polygons. Thus all opposite polygons are symmetric; but since polygons can slide around on the sphere, it follows that symmetric polygons are not necessarily opposite, although they are congruent to opposite polygons. Since the dihedral angles of the polyhedral angles have the same numerical measure as the angles of the spherical polygons, and the face angles of the former have the same numerical measure as the sides of the latter, it is evident that to each property of a polyhedral angle corresponds a reciprocal property of a spherical polygon, and vice versa. This relation appears by making the following substitutions : In addition to the correspondences between polyhedral angles and spherical polygons, it will be observed that a relation exists between a straight line in a plane and a greatcircle arc on a sphere. Thus, to a plane triangle corresponds a spherical triangle, to a straight line perpendicular to a straight line corresponds a great-circle arc perpendicular to a great-circle arc, etc. The word arc is always understood to mean great-circle arc, in the geometry of the sphere, unless the contrary is stated. It is very desirable that every school have a spherical blackboard. It is only by its use that students come to a clear knowledge of spherical geometry. If such a blackboard is at hand it is recommended that many problems and exercises of Book I be investigated on the sphere. ..... 9 DEFINITION. Two polyhedral angles V-ABC ....., V'-A'B'C' are said to be congruent when the first can be placed on the second so that V coincides with V', VA lies along V'A', VB lies along V'B', ..... The properties of spherical polygons given on p. 286 require only slight proof, and for convenience of reference they are all classified as one theorem. The student should give the proof where it is required, and show that the references apply in the other cases. Either the proposition relating to the polyhedral angle, or the one relating to the spherical polygon, may be proved first, as is most convenient. Thus it is better to prove a before a', b' before b, etc. EXERCISE. 726. State without proof the propositions concerning spherical polygons, corresponding to exs. 626, 631. Theorem 11. a. Opposite polyhedral angles have their corresponding dihedral angles and face angles respectively equal, but arranged in reverse order. VI, th. 25 b. Two opposite or two symmetric trihedral angles are congruent if each has two equal dihedral angles, or two equal face angles. Follows from b' c. Hence, if a trihedral angle has two dihedral angles equal to each other, the opposite face angles are equal. Follows from c' d. In any trihedral angle, in which each face angle is less than a straight angle, the sum of any two face angles is greater, and their difference less, than the third angle. VI, th. 26 and cor. 1 e. In any polyhedral angle, in which each face angle is less than a straight angle, any face angle is less than the sum of all the remaining face angles. VI, th. 26, cor. 2 f. In any convex polyhedral angle the sum of the face angles is less than a perigon. VI, th. 27 g. No face angle of a convex polyhedral angle is greater than a straight angle. Follows from g' a'. Opposite spherical poly. gons have their corresponding angles and sides respectively equal, but arranged in reverse order. Follows from a b'. Two opposite or two symmetric spherical triangles are congruent if each has two equal angles or two equal sides. Let the student give the proof c'. Hence, if a spherical triangle has two angles equal to each other, the opposite sides are equal. Let the student give the proof d. In any spherical triangle, in which each side is less than a semicircumference, the sum of any two sides is greater, and their difference less, than the third side. Follows from d e'. In any spherical polygon, in which each side is less than a semicircumference, any side is less than the sum of all the remaining sides. Follows from e f'. In any convex spherical polygon the sum of the sides is less than a circumference. Follows from f g'. No side of a convex spherical polygon is greater than a semicircumference. Def. convex spher. polyg., cor. Theorem 12. If two angles of a spherical triangle are unequal, the opposite sides are unequal, and the greater angle has the greater side opposite. Given a spherical triangle ABC, with Proof. 1. Suppose BD drawn making < DBA = A. a 1'. In a trihedral angle, according as one face angle is greater than, equal to, or less than another, the dihedral angle opposite that face angle is greater than, equal to, or less than the dihedral angle opposite the other. 2'. If a trihedral angle has its three face angles equal, it has also its three dihedral angles equal, and conversely. EXERCISES. 727. State and answer the questions concerning spherical polygons, corresponding to ex. 611. 728. Construct from some spherical surface, such as an orange skin, apple skin, or paper sphere, two triangles corresponding to the trihedral angles mentioned in ex. 614, and see if they are congruent. In referring to polar triangles ABC, A'B'C', the above arrangement of elements will always be intended. Also, in referring to symmetric spherical triangles, ABC, and A'B'C', it will always be understood that A = A', etc., and AB A'B', etc. The polar triangle of ABC is often called the polar of ABC. Theorem 13. If one spherical triangle is the polar of a second, then the second is also the polar of the first. Given a spherical triangle, ABC, and A'B'C' its polar. To prove that A ABC is the polar of ▲ A'B'C'. Proof. 1. In the above figure, suppose AC', AB', drawn. 2. ... B' is a pole of b, AB' is a quadrant. Th. 6, cor. 3 3. Similarly, AC' is a quadrant. 4. .. A is a pole of a'. Th. 7 5. Similarly, B and C are poles of b' and c', respectively. |