4. Problem.— To draw a line parallel to a given line AB through a given point C. Method I. (Fig. 26).—Instruments: ruler and compasses. Join C to any point D in AB. With centre D and radius DC describe an arc, cutting AB in E. With centre C and radius DC describe the arc DF. Method II. (Fig. 27). —Instruments : ruler and compasses. With any point D in AB as centre and DC as radius, describe a semicircle AECB. With centre A and radius BC cut the semicircle in E. Draw CE. The line CE is the parallel required. Method III. (Fig. 28).— Instruments: ruler, and a piece of wood with three straight edges, called a triangle. Place the ruler and triangle as shown in the figure. Slide the tiiangle along till its edge touches the point C; then draw a straight line along the edge of the triangle. This line will be parallel to AB because the edge of the triangle during the motion remains parallel to AB. 5. Draw a straight line; then draw a parallel line through a point not in the line by Method I. 6. The same exercise, using Method II. 7. The same exercise, using Method III. 8. Draw a parallel to a given line through a given point by Method I. Test the accuracy of your result by Method III. 9. Draw by Method III. six parallels as nearly equidistant as you can. 10. Draw a vertical line, mark five points on it, and draw through the points parallel lines by Method III. 11. Draw six parallel lines, two vertical, two horizontal, and two inclined. 12. Draw a three-sided figure, and then draw through each of its corners a line || to the opposite side. 13. Describe a circle, and draw two parallel chords. 14. Draw a straight line; then try to draw freehand a parallel to it. Begin by marking points which, as well as you can judge, must be in the required line. 15. Draw freehand a series of six parallel lines. 16. Hold two pencils (1) parallel; (2) so that they would intersect if prolonged; (3) so that they are not parallel, and also would not intersect. 17. Point out on the body given to you edges which are (1) parallel; (2) intersecting; (3) neither parallel nor intersecting. 18. The same exercise with the edges of the room. 19. When is a straight line parallel to a plane? A straight line is parallel to a plane when the line will never meet the plane however far they are both produced. 20. When are two planes parallel to each other? 21. Give an example of a line || to a plane; also of two parallel planes. Lesson 9. 1. What has a straight line besides direction? Besides direction, a straight line has length. As regards length, two straight lines are either equal or unequal. 2. Define equal and unequal straight lines. Two straight lines are equal if they can be so placed, one upon the other, that their ends coincide. If this cannot be done, the lines are unequal. 3. How is the equality of two lines expressed? The equality of two lines AB and CD is expressed thus: AB = CD. A— B This is called an equation, and is C D read, "AB is equal to CD." E p The sign = is the sign of equality. Fi«. 29. 4. How is the inequality of two lines expressed? The inequality of the two lines AB, EF is thus expressed: AB > EF, or EF< AB. These expressions are read, ilAB is greater than EF," and "EF is less than AB." 5. How is the equality of two lines tested? The lines are usually compared with a third line. Suppose I wish to test whether the lines AB and CD (Fig. 29) are equal, I open the dividers, place one point on A, and the other on B. This is called "taking the distance AB between the points of the dividers." Then, keeping the opening of the dividers unchanged, I place one point on C, and observe whether the other point will fall on D. If it does fall on D, I know that AB = CD. Here the third line with which AB and CD are compared is the distance between the points of the dividers. 6. Draw freehand from a point two lines as nearly equal as you can, then test their equality with the dividers. 7. The same exercise, one line to be horizontal, the other vertical. 8. Draw freehand a three-sided figure, with its sides as nearly equal as you can; test the equality with the dividers. 9. Draw four parallels exactly equal in length. 10. Draw a line equal to an edge of the body given you. 11. Draw lines a, b, c, d so that a = b, c ><z, d < b. 12. Read the following: m = n, AB < CD, x >y. 13. Draw four straight lines, a, b, c, d; then compare a with each of the other lines, writing the result with the proper sign. 14. Draw freehand two lines as nearly equal as you can, each longer than the greatest opening of your dividers. Can you test their equality by means of your dividers? 15. Compare different lines drawn between two points. A straight line is the shortest line between two points; hence the length of a straight line joining two points is taken as the distance of the points from each other. 16. How do sign-painters make use of this truth (Fig. 30)? They chalk a cord and stretch it tightly between the points through which the line is to pass; then, seizing this cord by the middle, they draw it back a little from the wood, and then let it go. It springs back, strikes the wood a sharp blow, and leaves on it a white trace, which is a straight line. 17. Define an axiom, and illustrate the meaning. There are statements or assertions which are so obvious that they stand in no need of any explanation or proof. Every one sees at a glance that they are true. In Geometry, an assertion which is admitted to be true, without proof, is called an axiom. An instance of an axiom occurs in No. 5. We assume that if AB and CD are separately equal to the distance between the points of the dividers, they are equal to each other. In general, if any two magnitudes are each equal to a third magnitude, we assume, as quite obvious, that they are equal to each other. Another example of an axiom is the assertion that a straight line is the shortest distance between two points. 18. State the most important axioms which are found useful in tJie study of Geometry. 1. Two magnitudes, each equal to a third, are equal to each other; or, for any magnitude, its equal may be substituted. 2. If equals are added to equals, the sums are equal. 3. If equals are taken from equals, the remainders are equal. 4. If equals are multiplied by equals, the products are equal. 5. If equals are divided by equals, the quotients are equal. 6. A whole is greater than any of its parts. - 7. Through a given point in a given direction, only one straight line can be drawn. 8. Through two points, only one straight line can be drawn. 9. A straight line is the shortest distance between two points. 10. Through a given point only one parallel to a given straight line can be drawn. |