CHAPTER VI STRAIGHT LINES AND PLANES 534. Surface. A boundary of a portion of space is called a surface. A surface may be limited or unlimited in extent. It has two dimensions. 535. Plane. A surface such that a straight line joining any two of its points, lies wholly in the surface, is called a plane surface or, simply, a plane. Since a straight line may be unlimited in length, this definition implies that a plane may be unlimited in extent, for otherwise the straight line could not lie wholly in the surface. It also follows from the definition that a straight line can intersect a plane in but one point. D M B 536. Representing a Plane. In drawing a geometric figure a portion of the plane is conveniently represented as a rectangle seen obliquely. In order to make the plane appear to "stand out," the front edge is frequently drawn longer than the back edge. Making the edges on the front and one end heavy and shading help one to image a plane as it is intended it should appear. 537. Reading a Plane. A plane may be read by a single letter, by letters at opposite vertices of the rectangle that represents it, or by the letters at all the vertices of the rectangle. Thus, in the figure, the planes may be read "plane M," "plane AC," or "plane ABCD.” DETERMINING A PLANE 538. A plane is said to be determined by given conditions if that plane and no other plane ful fills those conditions. 539. Axiom. Three points not in a straight line determine a plane. That is, one plane and only one plane contains three points that are not in a straight line. 540. Theorem. A straight line and a point not in that line determine a plane. Given the straight line MN and the A point P not in the line. M P B N R To prove that MN and P determine a plane. .. MN and P determine a plane. 541. Remark. Since a straight line and any point in space determine a plane, it follows that an indefinite number of planes can be passed through a straight line. $539 § 535 538 542. Theorem. Two intersecting straight lines determine a plane. S C One line and a point in the other determine a A plane. B D T Complete the proof. M 543. Theorem. Two parallel lines determine a plane. 544. Theorem. A straight line lies in a plane if it has a yoint in the plane and is parallel to a line in the plane. EXERCISES 1. Hold two pencils so that they determine a plane. So that they do not determine a plane. 2. In making a kite, two straight sticks AB and CD are fastened together at P. Show that paper stretched over these sticks lies in a plane. 3. Show that a rubber cord fastened at both ends and stretched by grasping it at any other point, lies in one plane. 4. From a given point lines are drawn to points on a given line; show that all these lines lie in the same plane. D P A B 5. What is the locus of all lines that pass through a given point and intersect a given line not containing the point? 6. Show that, if a rubber band is stretched by three hooks in any position, all parts of the band lie in one plane. 7. Do four points necessarily lie in a plane? When do they? Give illustrations by using points in the classroom. 8. Why is a tripod used to support a camera or a surveyor's transit? 9. Why do the four legs of a chair sometimes not all rest upon a floor? 10. How many planes are determined by four points not all lying in the same plane? 11. Given five points four of which are in the same plane. How many planes do they determine? 12. How many planes are determined by three A. concurrent lines that do not all lie in the same plane? 13. How does a carpenter determine whether a floor is a plane? B C 14. Why does a mason use a trowel with long straight edges when "truing up" a wall? 15. How many planes are determined by four lines all meeting in a point but no three of which lie in the same plane? 16. Must a triangle lie in a plane? Must a parallelogram? Must a trapezoid? Must every quadrilateral? 17. Is the following a complete definition for a circle: A curved line every point of which is equally distant from a fixed point called the center is a circle? Why? 18. Through a point in space (1) draw a line parallel to a given line; (2) draw a line perpendicular to a given line. RELATIVE POSITIONS OF LINES AND PLANES 545. Definitions. Collinear means lying in the same line. Coplanar means lying in the same plane. 546. Relations of Two Lines. From the study of plane geometry and §§ 542, 543, it follows: (1) That two coplanar lines may coincide, intersect, or be parallel. (2) That two non-coplanar lines are neither intersecting nor parallel. 547. Relations of a Line and a Plane. have three positions relative to a plane: (1) It may lie in the plane. (2) It may intersect the plane. (3) It may be parallel to the plane. A straight line may A line lies in a plane if all of its points are in common with the plane. A line intersects a plane if it has only one point in common with the plane. The point in common is called the foot of the line. A line is parallel to a plane, and the plane is parallel to the line, if they have no point in common, that is, if they do not meet. 548. Relations of Two Planes. Two planes may have three relative positions: (1) They may coincide. (2) They may intersect. (3) They may be parallel. Two planes are said to coincide if they have the same determining conditions. In plane geometry, the relative positions of points and lines are few and so the treatment of the subject is quite informal. In solid geometry, the study is made of points, lines, and planes, and their relative positions are many. It is advisable then, to make a more formal study of the subject. INTERSECTING PLANES 549. Definition. The intersection of two surfaces consists of all points common to the two surfaces, and no other points; that is, it is the locus of all points common to the two surfaces. 550. Axiom. If two planes have one point in common, they also have another point in common. 551. Theorem. If two planes intersect, their intersection is a straight line. Given two intersecting planes P and Q. To prove that their intersection is a straight line. Proof. Let A and B be two points common to both planes. $550 The straight line determined by A and B lies in the plane P and also in Q. § 535 Why? Then AB is common to the two planes. for then they would coincide. § 540 ..the intersection of planes P and Q is a straight line. EXERCISES 1. Can three planes intersect (1) in one line? (2) in two lines? (3) in three lines? (4) in more than three lines? Explain. 2. If a paper is folded and creased, why does it form a straightedge along the crease? 3. From a point external to two non-coplanar lines only one line can be drawn that will cut the two lines. |
