PROP. XXXV. THEOREM 408. If two triedral angles have the face angles of one equal respectively to the face angles of the other, their homologous diedral angles are equal. 444 FIG. 1. FIG. 2. Given, in triedral ▲ O-ABC and O'-A'B'C', FIG. 3. ZAOBLA'O'B,' / BOC = LB'O'C', To Prove diedral / OA = diedral O'A'. A' Proof. 1. Lay off OA, OB, OC, O'A', O'B', O'C' all equal; draw lines AB, BC, CA, A'B', B'C', C'A'. 2. We have A OAB = ▲ O'A'B' (§ 46); then AB = A'B'. (?) 3. In like manner, BC = B'C', CA C'A'. 4. Then ▲ ABC = A A'B'C' (§ 52); and EAF=E'A'F. (?) 5. Take AD = A'D'; and draw DE in face OAB 1 OA, meeting AB at E (it will meet, since ▲ OAB is isosceles, and ZOAB acute). 6. Draw DF in face OACL OA, meeting AC at F; and D'E' and D'F' in faces O'A'B', O'A'C', 1 O'A', meeting A'B' at E', and A'C' at F"; and draw lines EF, E'F'. 7. In rt. A ADE, A'D'E', AD = A'D'; and from equal A OAB, O'A'B', Z DAE = L D'A'E'. (?) 8. Then ▲ ADE = ▲ A'D'E' (§ 89); whence, AE = A'E', and DE = : D'E'. 9. In like manner, AF A'F", = DF: = D'F". 10. From results (4), (8) and (9), ▲ AEF=▲ A'E'F", and EFE'F'. 11. From results (8), (9) and (10), ▲ DEF = ^ D'E'F'. (§ 52) 12. Then, EDF = / E'D'F' (?); these are plane of diedrals OA and OA (§ 379), whence the diedrals are equal by § 383. The above proof holds for Fig. 3 as well as for Fig. 2; in Figs. 1 and 2, the equal parts occur in the same order, and in Figs. 1 and 3 in the reverse order. 409. It follows from § 408 that if two triedral angles have the face angles of one equal respectively to the face angles of the other, 1. They are equal if the equal parts occur in the same order. For if triedral Z O'-A'B'C' (Fig. 2) be applied to O-ABC so that diedral O'A' and OA coincide, point O' falling at 0, then since A'O'C' AOC, and Z A'O'B' = ZAOB, O'B' will coincide with OB, and O'C" with OC. = 2. They are symmetrical if the equal parts occur in the reverse order. BOOK VII POLYEDRONS DEFINITIONS 410. A polyedron is a solid bounded by polygons. We call the bounding polygons the faces of the polyedron, their sides the edges, and their vertices the vertices; a diagonal is a straight line joining two vertices not in the same face. The least number of polygons which can bound a polyedron is four. A polyedron of four faces is called a tetraedron; of six faces, a hexaedron; of eight faces, an octaedron; of twelve faces, a dodecaedron; of twenty faces, an icosaedron. 411. A polyedron is called convex when the section made by any plane is a convex polygon (§ 118). All polyedrons considered hereafter will be understood to be convex. 412. Any two solids are called equivalent when they have the same volume. PRISMS AND PARALLELOPIPEDS 413. A prism is a polyedron, two of whose faces are equal polygons lying in parallel planes, having their homologous sides parallel, the other faces being parallelograms (§ 107). We call the equal and parallel faces the bases of the prism, and the others the lateral faces; we call the edges which are not sides of the bases the lateral edges, and the sum of the areas of the lateral faces the lateral area. The altitude is the perpendicular distance between the planes of the bases. 414. The following is given for reference: The bases of a prism are equal. 415. It follows from 413 that the lateral edges of a prism are equal and parallel. (§ 104) 416. A prism is called triangular, quadrangular, etc., according as its base is a triangle, quadrilateral, etc. 417. A right prism is a prism whose lateral edges are perpendicular to its bases. The lateral faces are rectangles (§ 350). An oblique prism is a prism whose lateral edges are not perpendicular to its bases. 418. A regular prism is a right prism whose base is a regular polygon. 419. A truncated prism is a portion of a prism included between the base, and a plane, not parallel to the base, cutting all the lateral edges. The base of the prism and the section made by the plane are called the bases of the truncated prism. 420. A right section of a prism is a section made by a plane cutting all the lateral edges, and perpendicular to them. 421. A parallelopiped is a prism whose bases are parallelograms; that is, all the faces are parallelograms. A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to its bases. A rectangular parallelopiped is a right parallelopiped whose bases are rectangles; that is, all the faces are rectangles. cube is a rectangular parallelopiped who e six faces are all squares. Proof. 1. The homologous sides CD, C'D', etc., are . (§ 364) 2. Then, the homologous sides and are equal (§§ 105, 1; 376); then use § 123. PROP. II. THEOREM 423. Two prisms are equal when the faces including a triedral angle of one are equal respectively to the faces including a triedral angle of the other, and similarly placed. Given, in prisms AH and A'H', faces ABCDE, AG, and AL equal respectively to faces A'B'C'D'E', A'G', and A'L'; the equal parts being similarly placed. To Prove prism AH prism A'H'. = Proof. 1. Since EABLE'A'B', LEAFLE'A'F", FAB FA'B' (§ 48) triedral & A-BEF, A'-B'E'F" are equal. (§ 409, 1) = 2. Apply prisms so that A'B'C'D'E', A'G', and A'L' shall coincide with ABCDE, AG, and AL, respectively. 3. Then edges C'H', D'K' fall on CH, DK, and vertices H', K' on H, K. (§§ 69, 415) 4. Then faces B'H', C'K', D'L', H'L' coincide with BH, CK, DL, HL (§§ 346, 347); and prisms are equal. |