PROPOSITION XIV. THEOREM. 468. If a straight line be parallel to another straight line drawn in a plane, it is parallel to the plane. M E B C F N Let AC be parallel to the line B D in the plane M N. We are to prove A C to the plane M N. From A and C, any two 1 to A C, and A E and C F points in A C, draw A B and C D to the plane M N. (if two not in the same plane have their sides || and lying in the same direction, they are equal). Now since the points A and C, any two points in the line A C, are equally distant from the plane MN, all the points in AC are equally distant from the plane MN. § 432 .. A C is to the plane M N. Q. E. D. 469. If two straight lines be intersected by three parallel planes their corresponding segments are proportional. M A C Let A B and C D be intersected by the parallel planes MN, PQ, RS, in the points A, E, B, and C, F, D. EG is to BD, (the intersections of two || planes by a third plane are || lines). $465 ON DIHEDRAL ANGLES. 470. DEF. The amount of rotation which one of two intersecting planes must make about their intersection in order to coincide with the other plane is called the Dihedral angle of the planes. The Faces of a dihedral angle are the intersecting planes. The Edge of a dihedral angle is the intersection of its faces. The Plane angle of a dihedral angle is the plane angle formed by two straight lines, one in each plane, perpendicular to the edge at the same point. perpendicular to the edge A B at the same point P. 471. The plane angle of a dihedral angle has the same magnitude from whatever point in the edge we draw the perpendicu lars. For every pair of such angles have their sides respectively parallel (§ 65), and hence are equal (§ 462). Two equal dihedral angles, D-A B-C', and D-A B-E', have corresponding equal plane angles, DAC and DAE. This may be shown by superposi tion. Any two dihedral angles, C-A B-E' and E-A B-H', have the same ratio as their corresponding plane angles, CA E and E A H. This may be shown by the method employed in B $200 and $201. Hence a dihedral angle is measured by its plane angle. It must be observed that the sides of the A E H plane angle which measures the dihedral angle must be perpendicular to the edge. Thus in the rectangular solid A H, Fig. 1, the dihedral angle F-B A-H, is a right dihedral angle, and is measured by the angle CED, if its sides CE and ED, drawn in the planes AF and AG respectively, be perpendicular to A B. But angle C'E' D', drawn as represented in the diagram, is acute, while angle C" E" D", drawn as represented, is obtuse. Fig. 2. Many properties of dihedral angles can be established which are analogous to propositions relating to plane angles. Let the student prove the following: 1. If two planes intersect each other, their vertical dihedral angles are equal. 2. If a plane intersect two parallel planes, the exteriorinterior dihedral angles are equal; the alternate-interior dihedral angles are equal; the two interior dihedral angles on the same side of the secant plane are supplements of each other. 3. When two planes are cut by a third plane, if the exteriorinterior dihedral angles be equal, or the alternate dihedral angles be equal, or the two interior dihedral angles on the same side of the secant plane be supplements of each other, the two planes are parallel. 4. Two dihedral angles are equal if their faces be respectively parallel and lie in the same direction, or opposite directions, from the edges. 5. Two dihedral angles are supplements of each other if two of their faces be parallel and lie in the same direction, and the other faces be parallel and lie in the opposite direction, from the edges. PROPOSITION XVI. THEOREM. 472. If a straight line be perpendicular to a plane every plane embracing the line is perpendicular to that plane. Let A B be perpendicular to the plane M N. We are to prove any plane, PQ, embracing A B, perpendicular to MN. At B draw, in the plane MN, BC to the intersection DQ. (if a straight line be § 430 to a plane, it is to every straight line in that plane drawn through its foot). Now ABC is the measure of the dihedral P-D Q-N. But ABC is a right angle, .. the P-D Q-N is a right dihedral, ..PQ is to MN. § 470 Q. E. D. |