478. Every point in the plane that bisects a dihedral angle is equidistant from the faces of that angie. Given: P, a point in plane MA, bisecting dihedral angle CABD; To Prove: P is equidistant from AC and BD. From P draw PE to AC, and PF to BD; (429) through PE and PF pass a plane intersecting AC in OE, BD in OF, and therefore AM in OP. Since PE is to AC, and PF to BD, plane PEF is 1 to AB; (Const.) (476) .. POE and POF are the plane angles that measure the 479. DEFINITION. The projection of a point on a plane is the foot of the perpendicular from the point to the plane. 480. DEFINITION. The projection of a line on a plane is the locus of the projections on that plane of all the points in the line. PROPOSITION XXII. THEOREM. 481. The projection of a straight line on a plane is a straight line. Given: A straight line AB and a plane MN; To Prove: The projection of AB upon MN is a straight line. Through AB pass a plane AD to plane MN, which it intersects in CD. Since AD is to MN, (Const.) AD contains all the perpendiculars from points in AB upon MN; (473) .. the feet of all these perpendiculars must meet MN in CD. But CD is a straight line; .. the projection of AB on MN is a straight line. (426) Q.E.D. EXERCISE 661. In the diagram for Prop. XXI., how many degrees must there be in the plane angle of dihedral angle CABD, that FO may be equal to FP? 662. In the same diagram, how many degrees are there in that plane angle if FP is equal to the line joining FE ? 663. Prove that if a line is equal to its projection on a given plane, it is parallel to the plane. 664. Prove that parallel lines have their projections on the same plane in lines that are coincident or parallel. 665. In the diagram for Prop. XXII., show that if BA be produced to meet MN in E, E is a point in DC produced. 666. In the diagram for the preceding exercise, if BD: AC = m : n, what is the ratio of CE to CD? 482. The acute angle formed by a straight line with its own projection on a plane, is the least angle it makes with any line in that plane. Given: Angle ABC, formed by AB with its projection BC on plane MN; To Prove Angle ABC is less than angle ABD, formed by AB any other line in MN than BC. with Lay off BD = BC, and join AD. In ▲ ABC, ABD, AB = AB, BC= BD, (Const.) but AC <AD, (since AC is to plane MN;) :. Z ABC <≤ ABD. (432) (Hyp.) Q.E.D. (91) 483. DEFINITION. The acute angle formed by a straight line with its own projection on a plane is called the inclination of the line to the plane, or the angle of the line and plane. EXERCISE 667. Show that lines having their projections on the same plane, coincident or parallel, are not necessarily themselves parallel. 668. A line meets a plane obliquely; with what line in the plane does it make the greatest angle? 669. A line has an inclination of 42° to each of two intersecting planes; how many degrees are there in the plane angle of the dihedral angle formed by the planes ? PROPOSITION XXIV. THEOREM. 484. A common perpendicular can be drawn to any two given straight lines not in the same plane. B M α Given: AB and CD, two straight lines not in the same plane; To Prove: A common perpendicular can be drawn to AB and CD. Through CD pass a plane MN that is || to AB, (454) and let ab be the projection of AB upon MN. Since ab is to AB, ab is not to CD, (449) (since AB and CD cannot be parallel ;) (Hyp.) .. ab will meet CD, say in b. At b draw bB to ab in the projecting plane of AB. Since AB is to ab in the same plane, 485. COR. 1. Only one perpendicular can be drawn common to two straight lines not in the same plane. For if there could be two such perpendiculars, then, as can easily be shown, there could be two perpendiculars drawn in the same plane at the same point in a straight line, which is impossible (41). 486. COR. 2. The common perpendicular is the shortest distance between two straight lines not in the same plane. POLYHEDRAL ANGLES. B 487. A polyhedral or solid angle is the angle formed by three or more planes meeting in a common point. The point in which the planes meet is called the vertex; the intersections of the planes, the edges; and the portions of the planes bounded by the edges, the faces of the angle. Thus, in the polyhedral angle S-ABCDE, S is the vertex; SA, SB, etc., are the edges; and ASB, BSC, etc., are faces or face angles. It is to be noted that, in a polyhedral angle, every two adjacent edges form a face angle; and every two adjacent faces, a dihedral angle. It is also to be noted that the faces and edges of a polyhedral angle may be supposed to extend indefinitely. As a convenience in demonstration, however, portions of the faces and edges may be represented as cut off by a plane. The section formed by the intersection of the plane with the faces is a polygon, sometimes called the base of the polyhedral angle. 488. A polyhedral angle is convex if any section made by a plane cutting all its faces is a convex polygon; as ABCDE. It is to be understood that the polyhedral angles about to be treated of are convex. 489. A polyhedral angle is trihedral, tetrahedral, etc., according as it has three, four, etc., faces. 490. A trihedral anglé is rectangular, birectangular, or trirectangular, according as it has one, two, or three right dihedral angles. The ceiling and walls of a room form trirectangular angles. EXERCISE 670. In the diagram for Prop. XX., how many trihedral angles are represented, with what common vertex ? 671. In the same diagram, if SBQ is a right angle, of what class, according to Art. 490, is each of the four trihedral angles? |