INFORMAL PREPARATORY GEOMETRY 1. The adjoining figure is a cube. It has six surfaces. Each surface is bounded by four lines, straight lines. Each straight line is bounded by two points. The surfaces of a cube, which are smooth and flat, are called Plane Surfaces; they are such that a straightedge (ruler) will touch the surface at all points of the straightedge, no matter where the plane surface may be tested. 2. Plane Geometry is the study of figures like the square, the triangle, the circle, etc., figures which lie in a plane surface. A Plane Geometrical Figure is a combination of points and lines which lie in one plane surface. Only such figures are considered in plane geometry. Ex. 1. Test the surface of your desk with your ruler to determine whether the surface is a plane or not. (See § 1.) Ex. 2. What are some other objects which have plane surfaces ? Ex. 3. How do men who are laying a concrete walk make use of this test in order to make the surface of the walk approximately plane? 3. Solid Geometry is the study of figures like the cube, the sphere, the cylinder, the pyramid, the cone, etc. 4. A Point is represented to the eye by a small dot. A point is named by placing beside it a capital printed letter; as point A. A point represents position only. 5. A Straight Line is represented to the eye by a mark made by drawing a pencil, a pen, or a piece of crayon along the edge of a straightedge. A line represents length only. A Curved Line is a line no part of which is straight. A Broken Line is a line composed of different successive straight lines. F G 6. Lines like the adjoining ones are called closed lines. It is apparent that a closed line incloses a portion of the plane. 7. The word "line" will mean a straight line hereafter unless otherwise specified. Ex. 4. Place upon paper a single point. (a) Draw through it one straight line. (b) Can you draw through it another straight line? (c) A third? (d) How many straight lines can be drawn through one point? Ex. 5. Place upon paper a point A and a point B. (a) Draw from A to B a straight line. (b) What happens when you try to draw a second straight line from A to B? (c) How many different straight lines do you conclude can be drawn between two points? Ex. 6. Can more than one curved line be drawn between two points? Illustrate. Ex. 7. (a) When walking along a straight line, are you moving constantly in the same direction or not? (b) Answer the same question if you are walking along a curved line. Ex. 8. Draw a straight line 2 inches long. Extend it one inch in each direction. 8. It will be assumed as apparent from the preceding exer. cises that : (a) One and only one straight line can be drawn through two points. This fact is also expressed thus: two points determine a straight line. (b) A straight line can be extended indefinitely in each direction. 9. The straight line determined by points A and B is called the line AB. Ex. 9. Select three points which are not in one straight line. Letter them A, B, and C. taken two at a time. Ex. 10. If four towns are situated so that no three can be connected by one straight road, how many roads must be constructed if each town is to be connected with each of the others by a straight road? Illustrate by a drawing. Ex. 11. Draw the straight line determined by two points. Then turn the straightedge over, and again draw a straight line between the two points. If the edge is a true straightedge, the two straight lines will coincide (form one line). Why is this so ? Ex. 12. Make a straightedge by folding a piece of paper. Test it by the method suggested in the preceding exercise. Ex. 13. In order to walk across a field in a straight line, a boy selects two objects which are in the direction in which he wishes to go, one of them directly between him and the other. As he walks, he constantly keeps the first object between himself and the second. (a) Why can he guide himself in this manner? (b) What two points determine the straight line along which he walks? 10. Two lines, straight or curved, intersect if they have one or more common points. Intersection. B The common points are called Points of 11. Two straight lines can intersect at only one point. If they were to intersect in two points, there would be two straight lines through these two points, and this is impossible (§ 8). A B This fact is also stated thus: two intersecting straight lines determine a point. Ex. 14. Draw three straight lines intersecting by pairs which do not all pass through one point. How many points do they determine ? Ex. 15. If there are in a county four straight roads, each of which crosses each of the others, and no three of which meet at one point, how many crossings are there? Illustrate. Ex. 16. How many points are determined by five straight lines intersecting by pairs, no three of which pass through a common point? Ex. 17. Can you make any definite statement about the number of points of intersection of two curved lines? 12. A Line-segment or Segment is the part of a straight line between two points of the line; R as, segment RS. S 13. Two segments are equal if they can be placed so that the ends of the one are exactly upon the ends of the other. The tool for testing the equality of two segments is the dividers. The dividers are spread until the points are upon A and B respectively. If the dividers can then be placed with their points on C and D respectively without changing the position of the legs of the dividers, then the two segments are equal. A AB is less than (<) CD if AB equals a part of CD. Ex. 18. Determine by means of the dividers the relative lengths of AB and BC; of AB and CD; of AB and AD. Ex. 19. Draw any segment AB. On a line of B B D indefinite length, mark off from a point 0 of that line a segment equal to 2 AB; also one equal to 3 AB. Ex. 20. Draw segments AB and CD, with AB greater than CD. (a) On a line of indefinite length, mark off a segment equal to AB+ CD. (b) Mark off a segment equal to AB — CD. Ex. 21. Let AB and CD be two segments. Sup- A pose that AB is placed upon CD with point A on point C. (a) Where will B fall if AB = CD? (b) Where, if AB = } CD ? (c) Where, if AB is greater than CD? B C D Ex. 22. Suppose that two segments are each equal to a third segment. How do these two segments compare with each other? Ex. 23. Suppose that two segments are each equal to equal segHow do these segments compare with each other? ments. Ex. 24. Complete the following sentences: ... (a) If equal segments are added to equal segments, the sums are (b) If equal segments are subtracted from equal segments, the remainders are ... 14. It will be assumed as apparent that: the straight line-segment is the shortest line A‹ between two points. G The Distance between two points is the length of the segment of the straight line between the points. To obtain a straight line between two points, a carpenter stretches a piece of twine between the two points. In doing so, he assumes that the shortest line between two points is the straight line. Ex. 25. Why are streets usually made straight? Ex. 26. Why do people often "cut across a vacant corner lot? Ex. 27. Place upon paper points A, B, and C so that they do not all lie upon a straight line. Draw segments AB, BC, and AC. By means of your dividers compare the longest segment with the sum of the other two segments. 15. A point bisects a segment if it divides the segment into two equal segments. The point is called the Mid-point of the segment. A Thus, C bisects AB if AC = CB. |