Of Planes. Let the parallel planes MN and PA be intersected by the plane EH: then will the lines of intersection EF, GH, be parallel. For, if the lines EF, GH, were not parallel, they would meet each other if sufficiently produced, since they lie in the same plane. If this were so, the planes MN, PA, would M N meet each other, and, consequently, could not be parallel; which would be contrary to the supposition. THEOREM X. If two lines are parallel to a third line, though not in the same plane with it, they will be parallel to each other. At any point, G, in the line EF, let GI and GH be drawn in the planes FC, BE, and each perpendicular to FE at G Then, since the line EF is perpendicular to the lines GH GI, it will be perpendicular to the plane HGI (Th. iv). And since FE is perpendicular to the plane HGI, its parallels AB and DC will also be perpendicular to the same plane (Th. v). Hence, since the two lines AB, CD, are both perpendicular to the plane HGI, they will be parallel to each other Of Planes. THEOREM XI. If two angles, not situated in the same plane, have their sides parallel and lying in the same direction, the angles will ba equal. Let the angles ACE and Bdf have the sides AC parallel to BD, and CE to DF: then will the angle ACE be equal to the angle BDF. For, make AC equal to BD, and CE equal to DF, and join AB, CD, and EF; also, draw AE, BF. Now since AC is equal and parallel to BD, the figure AD will be a parallelogram (Bk. I. Th. xxv); therefore, AB is equal and parallel to CD. D B E Again, since CE is equal and parallel to DF, CF will be a parallelogram, and EF will be equal and parallel to CD. Then, since AB and EF are both parallel to CD, they will be parallel to each other (Th. x); and since they are each equal to CD, they will be equal to each other. Hence, the figure BAEF is a parallelogram (Bk. I. Th. xxv), and consequently, AE is equal to BF. Hence, the two triangles ACE and BDF have the three sides of the one equal to the three sides of the other, each to each, and therefore the angle ACE is equal to the angle BDF (Bk. I. Th. viii). THEOREM XII. If two planes are parallel, a straight line which is perpendicula; to the one will also be perpendicular to the other. Of Planes. Let MN and PQ be two parallel planes, and let AB be perpendicular to MN: then will it be perpendicular to PQ. For, draw any line, BC, in the plane PQ, and through the lines AB, BC, suppose the plane ABC to be drawn, intersecting M A C the plane MN in the line AD: then, the intersection AD will be parallel to BC (Th. ix). But since AB is perpendicular to the plane NM, it will be perpendicular to the straight line AD, and consequently, to its parallel BC (Bk. I. Th. xii. Cor.) In like manner, AB might be proved perpendicular to any other line of the plane PQ, which should pass through B; hence, it is perpendicular to the plane (Def. 1). Cor. It from any point as H, any oblique lines, as HEF, HDC, be drawn, the parallel planes will cut these lines proportionally. For, draw HAB perpendicular to the plane MN: then, by the theorem, it will also be perpendicular to PQ. Then draw AD, AE, BC, BF. Now, since AE, BF, are the intersections of the plane P M F H N FHB, with the two parallel planes MN, PQ, they are paral lel (Th ix.); and so also are AD, BC. HA : HB : HE: HF, Then, GEOMETRY. BOOK VI. OF SOLIDS. DEFINITIONS 1. Every solid bounded by planes is called a polyedron. 2. The planes which bound a polyedron are called faces. The straight lines in which the faces intersect each other, are called the edges of the polyedron, and the points at which the edges intersect, are called the vertices of the angles, or vertices of the polyedron. 3. Two polyedrons are similar, when they are contained by the same number of similar planes, and have their polyedral angles equal, each to each. 4. A prism is a solid, whose ends arc equal polygons, and whose side faces are parallelograms. Thus, the prism whose lower base is the pentagon ABCDE, terminates in an equal and parallel pentagon FGHIK, which is called the upper base. The side faces of the prism are the parallelograms DH, DK, EF, TH AG, and BH. These are called the convex, or lateral surface of the prin Of the Prism. 5. The altitude of a prism is the distance between its upper and lower bases: that is, it is a line drawn from a point of the upper base, perpendicular, to the lower base 6, A right prism is one in which the edges AF, BG, EK, HC, and DI, are perpendicular to the bases. In the right prism, either of the perpendicular edges is equal to the altitude. In the oblique prism the altitude is less than the edge. F K H 7. A prism whose base is a triangle, is called a triangular prism; if the base is a quadrangle, it is called a quadrangular prism; if a pentagon, a pentagonal prisın; if a hexagon a hexagonal prism; &c. 8 A prism whose base is a parallelogram, and all of whose faces are also parallelograms, is called a parallelopipedon. If all the faces are rectangles, it is called a rectangular parallelopipedon. 9. If the faces of the rectangular parallelopipedon are squares, the solid is called a cube: hence, the cube is a prism bounded by six equal squares |