SOLID GEOMETRY. - BOOK VI. PROPOSITION XXVII. THEOREM. 464. If two triedrals have the face angles of one equal respectively to the face angles of the other, I. They are equal if the equal parts occur in the same order. II. They are symmetrical if the equal parts occur in the reverse order. I. In the triedrals O-ABC and O'-A'B'C', let Lay off the six equal distances OA, OB, OC, O'A', O'B', and O'C'; and draw AB, BC, CA, A'B', B'C', and C'A'. Draw OD and O'D' perpendicular to ABC and A'B'C', respectively; also, draw AD and A'D'. The equal oblique lines OA, OB, and OC meet the plane ABC at equal distances from D. ($ 408.) Hence, D is the centre of the circumscribed circle of the triangle ABC; and similarly, D' is the centre of the cir cumscribed circle of A'B'C'. Now apply O'-'B'C' to O-ABC, so that the points A', B', and C' shall fall at A, B, and C, and the point D' at D. Then the perpendicular O'D' will fall upon OD. (§ 399.) But the right triangles OAD and O'A'D' are equal. = ($ 88.) Whence, O'D' OD, and the point O will fall at O. Therefore, the triedrals O-ABC and O'-A'B'C' coincide throughout, and are equal. II. In the triedrals O-ABC and O"-A"B" C", let the angles AOB, BOC, and COA be equal respectively to A" O'B", B"O"C", and C"O"A". To prove O-ABC symmetrical to O"-A"B" C". Construct O-A'B'C' symmetrical to O"-A"B"C", having the angles A'O'B', B'O'C', and C'O'A' equal respectively to A"O"B", B"O" C", and C"O"A". Then the triedrals O-ABC and O'-A'B'C' have the angles AOB, BOC, and COA equal respectively to A'O'B', B'O'C', and C'O'A'. = Hence, triedral O-ABC triedral O'-A'B'C'. (§ 464, I.) Therefore, O-ABC is symmetrical to O"-A"B"C". 465. COR. If two triedrals have the face angles of one equal respectively to the face angles of the other, their homologous diedrals are equal. EXERCISES. 24. Two triedrals are equal when two face angles and the included diedral of one are equal respectively to two face angles and the included diedral of the other, and similarly arranged. 25. Two triedrals are equal when a face angle and the adjacent diedrals of one are equal respectively to a face angle and the adjacent diedrals of the other, and similarly arranged. 26. A is any point in the face EG of the diedral DEFG. If AC be drawn perpendicular to the edge EF, and AB perpendicular to the face DF, prove that the plane determined by AC and BC is perpendicular to EF. 27. From any point E within the diedral CABD, EF and EG are drawn perpendicular to the faces ABC and ABD, and GI perpendicular to the face ABC at II. Prove FII perpendicular to AB. BOOK VII. POLYEDRONS. DEFINITIONS. 466. A polyedron is a solid bounded by planes. The bounding planes are called the faces of the polyedron; their intersections are called the edges, and the intersections of the edges the vertices. A diagonal is a straight line joining any two vertices not in the same face. 467. The least number of planes which can form a polyedral is three (§ 453); hence, the least number of planes 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. 468. A polyedron is called convex when the section made by any plane is a convex polygon (§ 120). All polyedrons considered hereafter will be understood to be convex. 469. The volume of a solid is its ratio to another solid, called the unit of volume, adopted arbitrarily as the unit of measure (§ 179). 470. Two solids are said to be equivalent when their volumes are equal. PRISMS AND PARALLELOPIPEDS. 471. 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. The equal and parallel faces are called the bases of the prism, and the remaining faces the lateral faces; the intersections of the lateral faces are called 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. 472. The following is given for convenience of reference: The bases of a prism are equal. 473. It follows from the definition of § 471 that The lateral edges of a prism are equal and parallel. 474. A prism is called triangular, quadrangular, etc., according as its base is a triangle, quadrilateral, etc. 475. A right prism is a prism whose lateral edges are perpendicular to its bases. An oblique prism is a prism whose lateral edges are not perpendicular to its bases. 476. A regular prism is a right prism whose base is a regular polygon. 477. A truncated prism is that portion of a prism included between the base, and a plane, not parallel to the base, cutting all the lateral edges. 478. A right section of a prism is the section made by a plane perpendicular to the lateral edges. 479. A parallelopiped is a prism whose bases are parallelograms; that is, all the faces are parallelograms. 480. A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to its bases. 481. A rectangular parallelopiped is a right parallelopiped whose bases are rectangles; that is, all the faces are rectangles. The dimensions are the three edges which meet at any vertex. 482. A cube is a rectangular parallelopiped whose six faces are all squares. PROPOSITION I. THEOREM. 483. The sections of a prism made by two parallel planes which cut all the lateral edges, are equal polygons. Let the parallel planes CF and C'F' cut all the lateral edges of the prism AB. To prove that the sections CDEFG and C'D'E'F'G' are equal. We have CD parallel to C'D', DE to D'E', etc. (§ 417.) = (§ 105.) |