line alone is used, it shall be understood as referring to a straight line. If a point divides a straight line into two parts, each part is called a ray. Thus point P divides line QR into the rays PR and PQ. Evidently, a ray has only one end, and extends away from that end indefinitely. The portion of a straight line lying between two points of it is called a line-segment or segment. Thus MN is a linesegment, with ends at Mand N. 9. Properties of straight lines. - Two straight lines that cross at a point are said to intersect. Two straight lines in a plane that do not meet at all are called parallel. INTERSECTING LINES PARALLEL LINES Mark a point. Draw four straight lines through it. How many straight lines can intersect at one point? Mark two points, A and B. Draw a straight line passing through both points. Can a second straight line be drawn through these two points? Mark a point. Draw two straight lines through it. Can these two lines intersect at a second point? ! What is the shortest distance between two points? : The properties of straight lines may be summarized as follows: I. Any number of straight lines can be drawn through a point. II. One and only one straight line can be drawn through two points. Or, two points determine a straight line. It follows that two straight lines which are placed so that they have two or more points in common form one line. III. Two straight lines cannot intersect in more than one point. Or, two intersecting straight lines determine a point. IV. The length of the line-segment connecting two points is the shortest distance between them. The segment is taken to represent the distance between the points. EXERCISES 1. Fold and crease a piece of paper to make a straightedge. 2. If the edge of your ruler is placed so that it touches a straight line at two points, will it touch the line all along the ruler? Upon what property of the straight line in § 9 is the answer based? 3. Test the accuracy of your ruler as follows: Mark along the edge. Then reverse the ruler, placing it so that it touches the mark at two points, and draw a second mark. What will show whether the ruler is true or not? 4. Test the straightness of the segment AB as follows: Place over it a piece of thin paper, through which the segment may be seen, and trace it, marking points A' and B' over A and B, respec A B tively. Then reverse the paper so that point A' falls upon point B and point B' upon point A. Is the segment straight? Explain. 5. Explain why agun or other sighting instrument has two "sights" on it. 6. Explain what is meant by "Two points determine a straight line." 7. Show how the principle in Ex. 6 may be used to set three stakes in a straight line. 8. Explain how the fact that when a cord is stretched taut it is straight, follows from the fourth property of a straight line in § 9. 9. Three points not in a straight line determine how many straight lines? Draw them. Four points no three of which are in a straight line determine how many straight lines? Draw them. 10. Explain what is meant by "Two intersecting straight lines determine a point." 11. How could you locate a point known to be on each of two straight lines? 12. Draw two parallel lines by marking along both edges of a ruler while holding it in one position. What is known about two straight lines which are not parallel? 13. Point out four pairs of parallel lines in the room. 10. Construction, measurement, and comparison of line-segments. -Two things are said to coincide when they are placed in identically the same position throughout. Two line-segments are equal if, and only if, they may be made to coincide throughout. Compasses may be used to mark off equal segments on a line, to construct a segment equal to a given segment, or to compare the lengths of seg 1. Draw a line and mark off a segment on it equal to a given line segment. 2. Mark off four equal segments on a line. 3. Draw a line-segment twice as long as a given line-segment. 4. Draw a line-segment five times as long as a given line-segment. 5. Draw a line-segment equal to the sum of two given line-segments. 6. Draw a line-segment equal to the sum of three given linesegments. 7. Draw a line-segment equal to the difference between two given line-segments. 8. Draw any two line-segments. Show by the compasses which is the longer. 9. Estimate which of the line-segments a and b is the longer. Check the conclusion by testing with compasses. b a C 10. Estimate which of the line-segments AB and CD is the shorter. Check the conclusion by use of compasses. ADB 11. To bisect a line-segment. - Any geometric magnitude is said to be bisected when it is cut into two equal parts. Any segment AB may be bisected as P follows: With the compasses conveniently opened, place one leg at A and with the other draw arcs on both sides of AB. Then, without changing the opening of the compasses, place one leg at B and with the other draw arcs cutting the first arcs at Pand Q. Draw the straight line PQ, intersecting AB at C. Then AB is bisected at C. Show that AC and CB are equal by testing them with compasses. EXERCISES 1. In drawing the arcs to bisect a given line-segment, how far must the compasses be opened? 2. Draw several different line-segments and bisect each. Test the accuracy of each construction by means of compasses. 3. Divide a given line-segment into four equal parts. (First bisect it, then bisect each of the parts.) 4. Divide a given line-segment into eight equal parts. 5. Can the method of bisecting a line-segment be employed to divide a given line-segment into five equal parts? Into six? Explain. Into how many equal parts in general can a line-segment be divided by this method? C 6. This drawing shows the design of a Gothic window. Arc BC is drawn with radius AB and center A. Arc AC is drawn with radius AB and center B. The small arcs are drawn with radii equal to AD and centers at A, D, and B. The center M of the circle is found by drawing arcs with centers A and B and radii equal to AF. Show how to locate the points D, E, and F. Draw a similar design upon any convenient line- A E D F B segment corresponding to AB. 7. Draw a design like the following. First divide a line-segment into eight equal parts. With the ends of these parts as centers, and with a radius equal to two of the parts, draw arcs as shown in the figure. In drawing the smaller arcs use the same centers. ANGLES 12. An angle. - The from the same point is sides of the angle and the point is called the vertex. Thus, in the angle here represented, BA and BCare the sides and point B the vertex. This angle may be named "angle ABC," or "angle B." When using the three letters to name the angle, note that the letter at the B vertex is written and read in the middle. figure formed by two rays drawn an angle. The rays are called the C 5 A An angle is sometimes named also by writing a figure or a small letter within it. Thus the above angle may be named "angle 5." In writing the name of an angle, the symbol ∠ is used in place of the word "angle." Thus "angle ABC" is written ∠ABC, and "angle a" is written ∠a. |