SOLID GEOMETRY BOOK VI LINES AND PLANES IN SPACE 421. The Nature of Solid Geometry. In plane geometry we deal with figures lying in a flat surface, studying their properties and relations and measuring the figures. In solid geometry we shall deal with figures not only of two dimensions but of three dimensions, also studying their properties and relations and measuring the figures. 422. Plane. A surface such that a straight line joining any two of its points lies wholly in the surface is called a plane. A plane is understood to be indefinite in extent, but it is conveniently represented by a rectangle seen obliquely, as here shown. 423. Determining a Plane. A plane is said to be determined by certain lines or points if it contains the given lines or points, and no other plane can contain them. When we suppose a plane to be drawn to include given points or lines, we are said to pass the plane through these points or lines. When a straight line is drawn from an external point to a plane, its point of contact with the plane is called its foot. 424. Intersection of Planes. The line that contains all the points common to two planes is called their intersection. 425. Postulate of Planes. Corresponding to the postulate that one straight line, and only one, can be drawn through two given points, the following postulate is assumed for planes : One plane, and only one, can be passed through two given intersecting straight lines. For it is apparent from the first figure that a plane may be made to turn about any single straight line AB, thus assuming different positions. But if CD intersects AB at P, as in the second figure, then when the plane through AB turns until it includes C, it must include D, since it includes two points, C and P, of the line (§ 422). If it turns any more, it will no longer contain C. 426. COROLLARY 1. A straight line and a point not in the line determine a plane. For example, line AB and point C in the above figure. 427. COROLLARY 2. Three points not in a straight line determine a plane. For by joining any one of them with the other two we have two intersecting lines (§ 425). 428. COROLLARY 3. Two parallel lines determine a plane. For two parallel lines lie in a plane (§ 93), and a plane containing either parallel and a point P in the other is determined (§ 426). PROPOSITION I. THEOREM 429. If two planes cut each other, their intersection is Given MN and PQ, two planes which cut each other. To prove that the planes MN and PQ intersect in a straight line. Proof. Let A and B be two points common to the two planes. Draw a straight line through the points A and B. (For it has two points in each plane.) § 422 No point not in the line AB can be in both planes; for one plane, and only one, can contain a straight line and a point without the line. § 426 Therefore the straight line through A and B contains all the points common to the two planes, and is consequently the intersection of the planes, by § 424. Q.E.D. Discussion. What is the corresponding statement in plane geometry? 430. Perpendicular to a Plane. If a straight line drawn to a plane is perpendicular to every straight line that passes through its foot and lies in the plane, it is said to be perpendicular to the plane. When a line is perpendicular to a plane, the plane is also said to be perpendicular to the line. PROPOSITION II. THEOREM 431. If a line is perpendicular to each of two other lines at their point of intersection, it is perpendicular to the plane of the two lines. M R N Given the line AO perpendicular to the lines OP and OR at O. To prove that AO is 1 to the plane MN of these lines. Proof. Through O draw in MN any other line OQ, and draw PR cutting OP, OQ, OR, at P, Q, and R. Produce AO to A', making OA' equal to OA, and join A and A' to each of the points P, Q, and R. Then OP and OR are each 1 to AA' at its mid-point. .. AP A'P, and AR = A'R. = .. ▲ APR is congruent to ▲ A'PR. :: LRPA = ▲ A'PR. QPA = A'PQ. That is, ..A PQA is congruent to ▲ PQA'. .. AO is § 150 8.80 § 67 § 68 § 151 .. AQ= A'Q (§ 67); and OQ is to AA' at O. Q.E. D. PROPOSITION III. THEOREM 432. All the perpendiculars that can be drawn to a given line at a given point lie in a plane which is perpendicular to the given line at the given point. Given the plane MN perpendicular to the line OY at O. To prove that OP, any line to OY at O, lies in MN. Proof. Let the plane containing OY and OP intersect the plane MN in the line OP'; then OY is to OP'. § 430 In the plane POY only one can be drawn to OY at O. § 57 Therefore OP and OP' coincide, and OP lies in MN. Hence every 1 to OY at O, as OQ, OR, lies in MN. Q. E.D. 433. COROLLARY 1. Through a given point in a given line one plane, and only one, can be passed perpendicular to the line. 434. COROLLARY 2. Through a given external point one plane, and only one, can be passed perpendicular to a given line. Given the line OY and the point P. Y Draw PO to OY, and OQ 1 to OY. Then OQ and OP determine a plane through P1 to OY. M --Q P N Only one such plane can be drawn; for only one can be drawn to OY from the point P (§ 82). 435. Oblique Line. A line that meets a plane but is not per pendicular to it is said to be oblique to the plane. |