629. Def.-A polyedron is convex when no face, if produced, will enter the polyedron. All the polyedrons treated of in this book will be understood to be convex. 630. Defs.-A prismatic surface is a surface composed of planes passed between each successive pair of a system of parallel lines. The parallel lines are called the edges of the prismatic surface. PROPOSITION I. THEOREM 631. The sections of a prismatic surface made by two parallel planes cutting its edges are equal polygons. GIVEN the prismatic surface AB cut by two parallel planes in the sections GHIKL and G'H'IK'L'. TO PROVE these polygons are equal. The sides GH, HI, etc., are parallel respectively to G'H', H'I', etc. Hence GH-G'H', HI=H'I'. § 544 $118 The polygons GHIKL and G'H'I'K'L' are therefore mutually equilateral and equiangular. Hence they can be made to coincide and are equal. Q. E. D. 632. COR. A prismatic surface can be generated by a straight line moving so as to remain always parallel to a fixed straight line (drawn parallel to the edges) and always cutting the perimeter of a section. Hint.-By plane geometry a straight line can move across each face remaining parallel to the lateral edges. 633. Defs.-A prism is a polyedron bounded by a prismatic surface and two parallel planes. PRISMS The equal sections of the prismatic surface formed by the parallel planes are called the bases of the prism; the portion of the prismatic surface between the bases consists of the lateral faces; the portions of the edges of the prismatic surface between the bases are the lateral edges of the prism. 634. Defs.-A right prism is one whose lateral edges are perpendicular to its bases. An oblique prism is one whose lateral edges are not perpendicular to its bases. 635. Def.-A regular prism is one whose bases are regular polygons and whose lateral edges are perpendicular to its bases. PROPOSITION II. THEOREM 636. The lateral faces of a prism are parallelograms. GIVEN TO PROVE the prism FS. its lateral faces are parallelograms. Consider the lateral face FQ. Its sides FP and HQ are parallel, being edges of the pris matic surface. $630 Also FH and PQ are parallel, being the intersections of two parallel planes with a third. Therefore FQ is a parallelogram. $544 114 Similarly the other lateral faces are proved to be parallelograms. Q. E. D. 637. COR. I. The lateral edges of a prism are equal. 638. COR. II. The lateral faces of a right prism are rectangles. 639. Def.-A parallelopiped is a prism whose bases are parallelograms. 640. Def.-A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to its bases. 641. Def.-A rectangular parallelopiped is a right parallelopiped whose bases are rectangles. 642. Def.-A cube is a right parallelopiped whose bases are squares and whose lateral edges are equal to the sides. of its base. 643. COR. III. All the faces of a parallelopiped are parallelograms. 644. COR. IV. All the faces of a rectangular parallelopiped are rectangles. 645. COR. V. All the faces of a cube are equal squares. 646. Any two opposite faces of a parallelopiped may be GIVEN-the parallelopiped AG, the bases being first taken as AC and EG. TO PROVE that any other two opposite faces, as AF and DG, may be taken as bases. The four lines AD, BC, FG, EH are parallel to each other. $114, 549 They may therefore be taken as the edges of a prismatic surface. $630 Also AB and AE are parallel to DC and DH respectively. $114 8551 Hence the planes AF and DG are parallel. having AF and DG as bases. 8633 Q. E. D. |