Proposition 11. Mutually Equiangular Triangles 254. Theorem. If two triangles on the same sphere or on equal spheres are mutually equiangular, they are mutually equilateral, and are either congruent or symmetric. Given T and T', two mutually equiangular A on the same sphere or on equal spheres. Prove that T and T' are mutually equilateral, and that they are either congruent or symmetric. Proof. Let AP be the polar A of AT, and let AP' be the polar Δ of ΔΤ'. Since AT and Tare mutually equiangular, Given the polar A Pand Pare mutually equilateral. S 200 Hence the polar A Pand P'are mutually equiangular. $ 253 Now AT and T are the polar of A Pand P'. $ 199 Hence AT and Tare mutually equilateral. $ 200 ..AT and Tare either congruent or symmetric. $ 253 The statement that mutually equiangular spherical triangles are mutually equilateral, and are either congruent or symmetric, is true only when they are on the same sphere or on equal spheres. When the spheres are unequal, the spherical triangles are unequal. Proposition 12. Isosceles Triangle 255. Theorem. In an isosceles spherical triangle the angles opposite the equal sides are equal; and conversely, if two angles of a spherical triangle are equal, the sides opposite these angles are equal. Fig. 1 Fig. 2 1. Given the isosceles spherical A ABC with AC=AB. Prove that ZB=ZC. Proof. Let AD be the arc of a great O which bisects ZA. Then AB=AC, Given AD= AD, Iden. and ZBAD=LCAD. Const. Hence ABDA and CDA are symmetric. $ 251 ..ZB=ZC. $ 204 2. Given the spherical AABC with ZB= 2C. Prove that AC=AB. Proof. Let A'B'C' be the polar A of A ABC. Then A'C'+ZB=180° and A'B'+ZC=180°. $ 200 .. AC'=A'B'. Axs. 5, 2 Then, by 1, ZB'=ZC'. .. AC=AB. $ 200, Ax. 5 Proposition 13. Unequal Parts 256. Theorem. If two angles of a spherical triangle are unequal, the sides opposite these angles are unequal, and the side opposite the greater angle is the greater; and conversely, if two sides are unequal, the angles opposite these sides are unequal, and the angle opposite the greater side is the greater. 1. Given the spherical AABC with <C>ZB. Prove that ABAC Proof. Let CD, the arc of a great O, make ZDCB=ZB. Then DB=DC. $ 255, 2 Now AD+DC >AC. $ 196 .. AD+DB > AC, or AB AC. Ax. 5 2. Given the spherical A ABC .with AB>AC. Prove that <C>ZB Proof. Using the indirect method, if ZC=ZB, then AB=AC($ 255, 2), which is impossible, since AB > AC; and if ZC<ZB, then, by 1, AB<AC, which is also impossible. ..<C>ZB. Exercises. Review 1. Find the volume of a truncated right triangular prism whose lateral edges are 1 in., 14 in., and 24 in. respectively, and whose base is an equilateral triangle 3 sq. in. in area. 2. The volume of any truncated triangular prism is one third the product of a right section and the sum of the lateral edges. 3. In two tetrahedrons which have a trihedral angle of one equal to a trihedral angle of the other, the edges of these angles are 2 in., 21 in., 3 in. in the first tetrahedron and 3 in., 31 in., 41 in. in the second. Find the ratio of the volumes of the tetrahedrons. 4. A polyhedron of eight faces and six vertices has how many edges? 5. Consider the possibility of a crystal with four faces and two edges; with six faces and four edges. 6. Consider the possibility of a four-edged polyhedron. 7. The volume of the first of two similar tetrahedrons is 32 cu. in., and to an edge 2 in. long in the first there corresponds an edge 2. in. long in the second. Find the volume of the second tetrahedron. 8. Given that the volumes of two similar polyhedrons are in the ratio 1:8, find the length of the edge of the first that corresponds to one 3 in. long in the second. 9. Find the volume of a prismoid in which the areas of the bases are 7 sq. in. and 4 sq. in. respectively, the area of the midsection is 5 sq. in., and the height is 8 in. 10. Show that the formula for the volume of a prismoid also applies to the cylinder and the cone. 11. Sketch the figure and state the proposition relating to trihedral angles that follows from each of $$ 251-255. 257. Spherical Segment. If a sphere is cut by two parallel planes, the solid thus formed between the planes is called a spherical segment. P А V=ith (3 r2 + 3 m+2+ ha), If one of the parallel planes is tangent to the sphere, the segment is called a spherical segment of one base. 258. Spherical Sector. The solid generated by the revolution of a sector of a circle about a diameter of the circle as an axis is called a spherical sector. Р In this figure the solid is generated by the revo А A lution of the sector AOB about the diameter PP'. BA B' The zone generated by the arc AB is called the base of the spherical sector. The formula for the volume V of such a solid is V=}rs, P' where S is the area of the base, and r is the radius of the sphere. 1 Exercises. Spherical Segments and Sectors 1. Find the volume of a spherical segment whose bases are 6 in. and 8 in. in diameter, and whose altitude is 2 in. 2. If the diameter of a sphere is 14 in. and the altitude of the zone forming the base of a spherical sector is 3 in., what is the volume of the sector? 3. By regarding a spherical segment of one base as the difference between a spherical sector (whose base is a zone of one base) and a cone, show that the formula for the volume is V= ğ Th’ (3 r – h), where r is the radius of the sphere and h is the altitude of the segment. |