13. Size of an angle. - The size of an angle does not depend upon the lengths of the marks representing its sides. The sides of an angle are rays, and hence extend without limit from the vertex. Thus ∠a is less than 2b, although their sides are represented by marks of equal lengths. C The size of an angle depends upon the relative directions of its sides. It may be measured by the amount of turning of one side about the ver- D tex required to make it fall upon the other side. Thus ∠BAD is larger than ∠BAС, because AB would have to be revolved about A farther to fall upon AD than upon AC. A B The sizes of two angles may be compared by placing one upon the other so that the vertices are together and a side of one falls along a side of the other, then observing the positions of the other sides. If neither angle may be moved, a trace or copy of one may be made upon thin paper, and this trace applied to the other. Two angles are equal if, and only if, they may be made to coincide throughout. (See § 10.) If an angle is placed upon an equal angle so that the vertices and a pair of sides coincide, the other sides must coincide. EXERCISE 1. Draw two angles. Name one ∠ABC and the other / DEF. Cut out or trace DEF and place it upon ∠ABC so that E falls upon B and ED falls along BA. If EF falls within ∠ABC, which angle is greater? What angle shows the difference? If EF falls along BC, what is known of the two angles? 14. To construct an angle equal to a given angle. - An angle may be drawn equal to a given angle as follows: A DB R QP Given 2 BAC. Draw OP. Now with the compasses opened conveniently, and with one leg at vertex A of the given angle, draw an arc cutting the sides of O the given angle at the points D and E, respectively. Now with one leg at O, and without changing the opening of the compasses, draw an arc cutting the ray OP at point Q. Then place one leg of the compasses at D and adjust the compasses until the other leg falls at E. Without changing the adjustment, place one leg at, and with the other draw an arc cutting the first arc at a point R. Draw OR. Then ∠POR = ∠ BAС. Test the equality of the angles by the method of § 13. 15. Addition, subtraction, and multiplication of angles. - Two angles having the same vertex and a common side between them are called adjacent angles. Thus 21 and 2 are adjacent. The sum of two angles is obtained by placing them adjacent. 2 1 The sum of two angles may be constructed as illustrated in the figure, by using the construction of an angle equal to a given angle shown in § 14. CAD is Here ∠BAC is drawn equal to ∠a. Then drawn equal to ∠b. Hence ∠ BAD = ∠ a + ∠b. tinuing the process three or more angles may be added. Also angles may be subtracted, or an angle may be multiplied, by means of the construction in § 14. EXERCISES 1. Draw any angle. Construct an angle equal to it. How may the construction be tested to see if it is accurate? 2. Draw any two angles. Construct an angle equal to their sum. 3. Draw any three angles, and construct their sum. 4. Draw two angles, one apparently larger than the other. Construct an angle equal to their difference. 5. Draw an angle. Then construct another angle twice as large. 6. Draw an angle. Then construct another angle three times as large. 7. An angle may be constructed equal to a given angle by use of a piece of paper with one straight edge, as follows: To construct an angle equal to ∠ABC, place the paper with the straight edge along AB, and mark the vertex B on the edge of the paper. Also mark the point N on the opposite edge of the MA paper where the side BC intersects it. Then_ZMBN=∠ABC. By using E C 16. To bisect an angle. An angle ABC may be bisected (See § 11) as follows: With center at Band any convenient radius, draw arcs cutting AB at Dand BC at E. Then with centers at D and E, and equal radii of any convenient length, draw arcs meeting at a point F. Draw the ray from B through F. Then ∠ABC is bisected by BF. B Test ∠ ABF and ∠ FBC to see if they are equal. A EXERCISES 1. Draw an angle and bisect it. Test to see if the construction is accurate. Repeat until proficient in the construction, and the process is well fixed in mind. 2. Divide a given angle into four equal parts. 3. Divide a given angle into eight equal parts. 4. Can the process of bisecting an angle be used to divide an angle Into how many equal parts can an into five equal parts? Into six? angle be divided by this method? 5. Draw two intersecting lines. Bisect each of the four angles thus formed. What is observed about these bisectors? 17. Classification of angles. - An angle whose sides extend in opposite directions so that they form one straight line is called a straight angle, as ∠ABC. It is evident that the entire angular magnitude around a point is the sum of two straight angles. It is called perigon. a C B A A right angle is one half of a straight angle, as ∠a or ∠b. It follows that a right angle may be constructed by bisecting a straight angle. Explain the process. ba An acute angle is an angle less than a right angle, as ∠k An obtuse angle is an angle greater than a right angle but less than a straight angle, as ∠m. A reflex angle is an angle greater than a straight angle but less than two straight angles, as ∠ n. 18. The measurement of angles. - If a right angle is divided into 90 equal parts, each of these parts is called a degree, which is the principal unit of meas ure of angles. Thus, a right angle contains 90 degrees, written 90°. A straight angle contains 180°. A perigon, or the entire angular magnitude around a point, contains 360°. A degree is divided into 60 minutes (written 60'), and a minute into 60 A seconds (written 60"). 90 C A QUADRANT B The quadrant and the protractor are instruments used for measuring the number of degrees in an angle. To use either instrument for measuring ∠BAC, the center is placed over the vertex A of the angle, and the instrument turned until the zero mark of the scale falls on the side AB. Then the point where the side AC crosses the scale shows the number of degrees in the angle. 1. Draw any angle. Then measure the number of degrees in it with a protractor. Explain the method. 2. Measure LABC in § 16 with the protractor. Measure ∠ABF and 2 FBC in the same figure. Are the latter equal? 3. Draw any angle and bisect it as in § 16. Test the accuracy of the construction by means of the protractor. |