EXERCISES 1169 If two parallel lines are not perpendicular to a plane, their projections upon that plane are parallel lines. 1170 Are two lines necessarily parallel whose projections upon a plane are parallel? 1171 If a straight line intersects two parallel planes, it makes equal angles with the planes. 1172 When will the projection of a circle upon a plane be a circle? When will the projection be a straight line? 1173 If the projections of n points upon a plane are in a straight line, then points are all in the same plane. 1174 If a line meets a plane obliquely, with what line in the plane does it make the greatest angle? 1175 A plane can be perpendicular to only one edge of a polyedral angle. 1176 A plane can be perpendicular to only two faces of a polyedral angle. 1177 If two face angles of a triedral angle are equal, the opposite diedral angles are equal. 1178 The three planes bisecting the three diedral angles of a triedral angle intersect in the same line. 1179 If the three face angles of a triedral angle are equal, the three diedral angles are equal. 1180 The three planes passed through the edges of a triedral angle, perpendicular to the opposite faces, intersect in the same line. 1181 Find the locus of a point equidistant from two intersecting planes. 1182 Find the locus of a point equidistant from the vertices of a triangle; of a rectangle. 1183 Find the locus of a point equidistant from the edges of a triedral angle. 1184 Find the locus of a point equidistant from the faces of a triedral angle. [Let AO be the intersection of the three planes bisecting the three diedral angles of the triedral angle (Ex. 1178). Then all points in AO are equidistant from the three faces (§ 570). out at least two of the angle-bisecting equidistant from the three faces of the the locus required.] Any point without AO is withplanes, and is therefore not triedral angle. Hence AO is BOOK VII POLYEDRONS, CYLINDERS, AND CONES POLYEDRONS DEFINITIONS 594 A polyedron is a solid bounded by planes. The faces of a polyedron are the polygons bounding it. 595 A diagonal of a polyedron is a straight line joining any two of its vertices not in the same face. 596 A convex polyedron is a polyedron every section of which is a convex polygon. All polyedrons considered in this work are convex. Tetraedron Hexaedron Octaedron Dodecaedron Icosaedron 597 A tetraedron is a polyedron of four faces. PRISMS 598 A prism is a polyedron two of whose faces are equal and parallel polygons, and whose other faces are parallelograms. Prisms 599 The bases of a prism are two equal parallel faces; the lateral faces are all the faces except the bases; the lateral edges are the intersections of the lateral faces; the base edges are the intersections of the bases with the lateral faces; and the lateral area is the sum of the areas of the lateral faces. 600 Prisms are triangular, quadrangular, etc., according as their bases are triangles, quadrilaterals, etc. 601 The altitude of a prism is the perpendicular between the planes of its bases. 602 A right prism is a prism whose lateral edges are perpendicular to its bases. An oblique prism is a prism whose lateral edges are oblique to its bases. 603 A regular prism is a right prism whose bases are regular polygons. 604 A truncated prism is a portion of a prism included between a base and a section formed by a plane oblique to the base and cutting all the lateral edges. 605 COROLLARY. The lateral edges of a prism are equal and parallel; the lateral edges of a right prism are equal to its altitude; and the lateral faces of a right prism are rectangles. 606 A parallelopiped is a prism whose bases are parallelograms. Parallelopipeds 607 An oblique parallelopiped is a parallelopiped whose lateral edges are oblique to its bases. 608 A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to its bases. 609 A rectangular parallelopiped is a right parallelopiped whose bases are rectangles. 610 A cube is a parallelopiped all of whose faces are squares. 611 A right section of a prism is a section formed by a plane perpendicular to its lateral edges. 612 The volume of a solid is the number of units of volume which it contains. In applied Geometry the unit of volume. is usually a cube whose edge is some linear unit; as, a cubic inch, a cubic foot, etc. 613 Two solids having equal volumes are equivalent. PROPOSITION I. THEOREM 614 Sections of a prism made by parallel planes cutting all the lateral edges are equal polygons. HYPOTHESIS. ABCDE and A'B'C'D'E' are sections of the prism MN, formed by parallel planes cutting all the lateral edges. CONCLUSION. ABCDE and A'B'C'D'E' are equal polygons. PROOF AB is || to A'B', BC is to B'C', etc. § 543 ..Z ABC = A'B'C', / BCD = B'C'D', etc. § 549 § 202 Also, AB = A'B', BC = B'C', etc. .. ABCDE and A'B'C'D'E' are mutually equilateral and equiangular and can be made to coincide. ·. ABCDE = A'B'C'D'E'. $89 Q. E. D. 615 COROLLARY 1. Any section of a prism made by a plane. parallel to the base is equal to the base. 616 COROLLARY 2. All right sections of a prism are equal. EXERCISES 1185 Any section of a prism made by a plane parallel to a lateral edge is a parallelogram. 1186 In the above figure, what kind of solid is MD' ? |