Proposition XIV. THEOREM. 468. If a straight line be parallel to another straight line drawn in a plane, it is parallel to the plane. Let A C be parallel to the line B D in the plane M N. We are to prove A C Il to the plane M N. From A and C, any two points in A C, draw A B and CD I to A C, and A E and C F I to the plane M N. Join B E and D F. § 65 § 135 (Il lines comprehended between II lines are equal), and A E is II to C F, § 458 (two-straight lines I to the same plane are II). iiZBA E= DCF, (if two As not in the same plane have their sides II and lying in the same direction, they are equal). $ 110 (two rt. A are equal when an acute 2 and the hypotenuse of the one are equal respectively to an acute and the hypotenuse of the other). A E=CF, (being homologous sides of equal A). 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. :: A C is Il to the plane M N. $ 432 Q. E. D. $ 462 PROPOSITION XV. THEOREM. 469. If two straight lines be intersected by three parallel planes their corresponding segments are proportional. м 2 Dis Let A B and C D be intersected by the parallel planes A E CF EB FD Join E G and FG. § 465 (the intersections of two || planes by a third plane are Il lines). • A E A G $ 275 (a line drawn through two sides of a All to the third side divides those sides proportionally). § 465 CF AG $ 275 FD = GD 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 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. Thus, in the diagram, C-A B-D is a dihedral angle, C B and D A are its faces, A B is its edge, O PH is its plane angle if OP H and H P in the faces be 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 perpendiculars. 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, D A C and D A E. This may be shown by superposition. Any two dihedral angles, C-A B-E' and E-A B-H', have the same ratio as their corresponding plane angles, C A E and E A H. This may be shown by the method employed in Bl § 200 and $201. Hence a dihedral angle is measured by its plane angle. It must be observed that the sides of the 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. C E D, if its sides C E and E D, drawn in the planes A F and A G 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. 19 4. Two dihedral angles are equal if their faces he 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. brose to ļ the caus Musnice ... PROPOSITION XVI. THEOREM. 472. If a straight line be perpendicular to a plane every plane embracing the line is perpendicular to that plane. N Let A B be perpendicular to the plane M N. We are to prove any plane, PQ, embracing A B, perpendicular to M N. At B draw, in the plane M N, BCI to the intersection DQ. Since A B is I to M N, it is I to DQ and BC, $ 430 (if a straight line be I to a plane, it is I to every straight line in that plane drawn through its foot). Now Z A B C is the measure of the dihedral ZP-D Q-N. § 470 But Z A B C is a right angle, .. the Z P-D Q-N is a right dihedral, ..PQ is I to M N. Q. E. D. |