An axiom is a statement about quantities in general which is assumed to be true without proof. The student is already familiar with a number of axioms used in algebra. The following axioms are given here for future reference: I. Things which are equal to the same thing, or to equal things, are equal to each other. II. If equals are added to equals, the sums are equal. III. If equals are subtracted from equals, the remainders are equal. IV. If equals are multiplied by equals, the products are equal. V. If equals are divided by equals, the quotients are equal. VI. Like powers, or like positive roots, of equals are equal. VII. If equals are added to or subtracted from unequals, or if unequals are multiplied or divided by the same positive number, the results are unequal in the same order. VIII. If unequals are subtracted from equals, the remainders are unequal in the reverse order. IX. If unequals are added to unequals in the same order, the sums are unequal in that order. X. The whole of a thing is equal to the sum of all of its parts, and is greater than any one of its parts. XI. If the first of three things is greater than the second, and the second greater than the third, then the first is greater than the third. XII. A quantity may be substituted for its equal in any expression. PERPENDICULAR LINES 22. Perpendicular lines. If two intersecting straight lines form right angles, the lines are perpendicular to each other. When one line is drawn perpendicular to another, the point of intersection is called the foot of the perpendicular. In written work the symbol often is used in place of the words "perpendicular," "perpendicular to," or "is perpendicular to." Thus in the figure of the next section, CD 1 AB. The length of the perpendicular line-segment from an external point to a line is called the distance from the point to the line. * 23. To draw a perpendicular to a given line at a given point.Since perpendicular lines meet at right angles, the process of drawing a line CD perpendicular to a given line AB at a given point 0 on AB is the same as the process of constructing a right angle BOC. Explain the complete process. B 24. To draw a perpendicular to a given line, through a given external point. In order to draw a line CD perpendicular to a given line AB, through a given point not on AB, proceed as follows: With any convenient radius and center at O, draw arcs intersecting AB at two points. With these two points as centers and equal radii of any convenient length, A draw two arcs that intersect. Draw a line through B and this point of intersection. This will be the required line. Point out the steps of the construction in the adjoining diagram. EXERCISES 1. Observe that in each of the constructions in § 23 and § 24, one point of the required perpendicular line CD was given, and only one other point was located through which to draw it. Of what property of the straight line in § 9 is this an application? 2. In the construction of § 24, to draw the arcs with center what length of radius must be taken? 3. Mark points on three different lines, and draw perpendiculars to the lines at those points. Test the accuracy with a protractor. 4. Draw a perpendicular to a given line, through a given external point. Repeat until the process is thoroughly fixed in mind. Test the accuracy with a protractor. 5. Mark two points on a straight line. Then draw a perpendicular to the line at each point. What is the relation of the two perpendiculars? 6. Draw a diameter of a circle. Draw a second diameter perpendicular to the first. 7. Study the constructions of the following designs, and see how the method of drawing perpendicular diameters in Ex. 6 is employed in their constructions. Draw designs similar to these, but larger. CHAPTER II PROOFS: PARALLEL AND PERPENDICULAR LINES 25. A transversal of lines. A straight line which intersects two or more straight lines is called a transversal of those lines. Thus EF is a transversal of AB and CD in the figure. A transversal of two lines makes four angles at each intersection, as 1, 2, Z3, Z4, and ≤ 5, ≤ 6, ≤7, and ≤ 8. E A F ≤ 3, 24, ≤ 5, and ≤ 6 are called interior angles, and 1, ≤ 2, 27, and 8 are called exterior angles. Two angles such as 1 and 25 are called corresponding angles. How many pairs of corresponding angles are there? Name them. Two angles such as ≤ 3 and ≤5 are called alternate interior angles. How many pairs of alternate interior angles are there? Name them. Two angles such as 2 and 8 are called alternate exterior angles. How many pairs of alternate exterior angles are there? Name them. Two angles such as 4 and 5 are called consecutive interior angles. Name another pair of consecutive interior angles. Two angles such as ≤1 and 8 are called consecutive exterior angles. Name another pair of consecutive exterior angles. EXERCISES 1. In the figure of § 25, what angles are equal? Why? 2. What angles in the same figure are supplementary? 26. Corresponding angles compared. By marking along both edges of a ruler, draw two parallel lines. Draw any transversal of these lines. Compare any two of the corresponding angles by tracing one on thin paper and applying the trace to the other, by the method suggested in Ex. 7, § 15, or by measuring the angles carefully with a protractor. Repeat the comparison by drawing a new figure. What is the conclusion? Hence we infer the following general truth: If two parallel lines are cut by a transversal, the corresponding angles are equal.* 27. Establishing truth by reasoning. — In the preceding pages a number of geometric figures have been drawn and the accuracy of the constructions tested by measurement, tracing, etc. Also it has been seen how a geometric truth, such as that in § 26 above, may be discovered and proved by means of measurements and various kinds of practical tests. This, however, is a slow, and often unsafe, method of testing the accuracy of the constructions of figures or of discovering or establishing general truths. It is better, if possible, to arrive at mathematical truth through a a course of reasoning alone. In the subsequent sections we shall see how this is done. The following exercises show the danger of drawing conclusions from the mere appearances of geometric figures. EXERCISES 1. Review Ex. 9 and Ex. 10, page 9. 2. Judge whether AB and CD in this figure are both straight lines. Then test each line by applying a ruler to it. *TO THE TEACHER. -See Suggestions to Teachers, page vii. B |