SOLID GEOMETRY BOOK VI LINES AND PLANES IN SPACE — POLYHEDRAL ANGLES 477. DEF. Geometry of space or solid geometry treats of fig ures whose elements are not all in the same plane. (17) 478. DEF. A plane is a surface such that a straight line joining any two points in it lies entirely in the surface. (14) A plane is determined by given points or lines, if one plane and only one, can be drawn through these points or lines. PROPOSITION I. THEOREM 479. A plane is determined : (1) By a straight line and a point without the line. (2) By three points not in the same straight line. (3) By two intersecting straight lines. (4) By two parallel straight lines. (1) To prove that a plane is determined by a given straight line AB and a given point C. Turn any plane El passing through AB about AB as an axis until it contains C. If the plane, so obtained, be turned in either direction, it would no longer contain C. Hence, the plane is determined by AB and C. (2) To prove that three given points A, B, and a determine a plane. Draw AB. Then AB and C determine the plane. (Case 1.) (3) To prove that two intersecting straight lines AB and AC 480. COR. The intersection of two planes is a straight line. For the intersection cannot contain three points not in a straight line, since only one plane can be passed through three such points. NOTE. Constructions in solid geometry are usually treated as if we possessed the physical means of making a plane that is determined by any of the above elements. Obviously this could be done only if we employed models. With ruler and compasses the work is largely symbolic; i.e. we obtain pictures which are symbols of the figures to be constructed, but we do not always obtain the required figures themselves, as we do in plane geometry. Ex. 1438. Why is a photographer's camera or a surveyor's transit supported by three legs? Ex. 1439. How many planes are determined by four points, which do not all lie in one plane? 481. DEF. A straight line and a plane are parallel if they do not meet, however far they may be produced. 482. DEF. Two planes are parallel if they do not meet, however far they may be produced. PROPOSITION II. THEOREM 483. The intersections of two parallel planes with a third plane are parallel. III Given the parallel planes I and II, intersected by a third plane III in AB and CD, respectively. Proof. AB and CD cannot meet, for otherwise planes I and II would meet. Hence AB and CD lie in the same plane. ABCD. Q. E. D. 484. Cor. Parallel lines included between parallel planes are equal. NOTE. The student should note that in order to prove the parallelism of two lines in solid geometry it is not sufficient to show that the lines cannot meet. It must be demonstrated also that the two lines lie in one plane. PROPOSITION III. THEOREM 485. A plane containing one, and only one, of two parallel lines is parallel to the other line. Given AB || CD, and plane MN containing CD only. Proof. AB and CD determine a plane, intersecting MN in CD. Hence, if AB meets MN, it must meet MN in CD. But, since AB || CD, this is impossible. plane MN || AB. Q. E. D. 486. REMARK. Proposition III furnishes a method for constructing a plane parallel to a given line, AB. In such a construction always construct first a line, CD, parallel to the given line AB, and then pass a plane through CD. Since an infinite number of planes can be passed through CD, it is obvious that this problem is indeterminate unless the required plane must also satisfy other conditions, e.g. to be parallel to a second line or to pass through a given point, or to pass through another line, etc. 487. COR. 1. Through a given straight line AB a plane can be passed parallel to any other given straight line CD; and if the lines are not parallel, only one such plane can be drawn. 488. COR. 2. Through a given point Pa plane can be passed parallel to any two given straight lines in space; and A A B C B 77 D if the lines are not parallel, only one such plane can be drawn. Εχ. 1440. Τo construct a plane parallel to a given line and passing through two given points. PROPOSITION IV. THEOREM 489. A line parallel to a plane is parallel to the intersection of this plane with any plane passing through the line. Given AB || plane MN, and plane BC containing AB, and intersecting MN in CD. Proof. AB and CD cannot meet, for otherwise AB would meet the plane MN. AB and CD are in the same plane. Hence ABCD. Q. E. D. 490. COR. 1. If each of two intersecting lines is parallel to a given plane, their plane is parallel to the given plane. |