Z 3. Locate perpendicular lines in the classroom; in Fig. 112. 4. Prove by means of an equation that X T y if two lines are perpendicular to each FIG. 112 other the two adjacent angles are right angles (Fig. 113). 5. Exercise 4 suggests the follow ing use of a triangle having one right xx angle for drawing a line perpenFIG. 113 dicular to a given line to AB (Fig. 114) and passing through a given point C. Directions: Place a right triangle XYZ so that one side of the right y y angle lies on AB. Slide the triangle FIG. 114 along AB until Y Z passes through C. Then draw a line along YZ. This is the required line. B m 61. Supplementary angles. Measure angles m and n (Fig. 115) and find the sum. Two angles whose sum is 180°, or a straight angle, are supplementary Fig. 115 angles. Each is said to be the supplement of the other. When two supplementary angles Fig. 116 are adjacent (Fig. 116) they are called adjacent supplementary angles. m n п EXERCISES 1. Using only a ruler, draw a sketch of two supplementary angles. Measure with a protractor each of the angles and test the accuracy of your drawing. 2. How many degrees are there in the supplement of 45°? of 20°? of 160°? of 130°? Tell how the supplement is found in each 67° 15', case. 3. Find the supplement of 67° 15'. Solution: 180o = 179° 60'. Subtracting we have the supplement=112° 45'. 4. Find the supplement of 110° 30'; of 25° 40'; of 18° 57'; of 90° 40' 32", arranging your work as in Exercise 3. 5. Find the supplement of a°; of x°. Write the result in the form of binomials. 6. Make a formula for finding the supplement, s, of any given angle, a. 7. State by an equation that ao and lo are supplementary. 8. State by equations that the following pairs of angles are supplementary xo and 50°; x° and įxo; (x+20)° and (2x — 4)o. 9. One of two supplementary angles is 5 times as large as the other. Find the two angles by means of an equation. 10. One of two supplementary angles is .8 as large as the other. Find the two angles. 11. Find two supplementary angles, one of which is 14 times as large as the other. 62. Complementary angles. Measure angles x and y (Fig. 117) and find the sum. Two angles whose sum is 90o, or a right angle, are complementary angles. Each is the complement of the other. Fig. 117 EXERCISES 1. Make a sketch of two complementary angles. Test your drawing with a protractor. 2. Make a sketch of two adjacent complementary angles. 3. Arranging your work as in Exercise 3 (861) find the complement of 30°; 70°; 80.5°; 27° 14'; 18° 25'; 16° 13' 40"'; 65° 25' 32". 4. Find the complement of a°; x°. Write the results as binomials. 5. Make a formula for finding the complement, c, of a given angle a. 6. State by means of an equation that 20° is the complement of 5.ro ao; that ro is the complement of 4 7. One of two complementary angles is 8 times as large as the other. Find the angles by using an equation. 8. A right angle is to be divided into two parts so that one is 52 times as large as the other. Find the two parts. 9. Draw a right triangle (a triangle having a right angle). Show by measuring that the acute angles are complementary. Naming the acute angles a and b, state the equation. 10. Find the acute angles of a right triangle if one is 3 times the other; s as large as the other. Make sketches of the angles. 63. Opposite angles. Draw two intersecting lines, as AB and CD (Fig. 118). Measure the angles m, n, r, and s. In each anglé write the number of degrees it contains. How do angles m B and r compare as to size? Com pare angles n and s. C State by means of equations the relations between m and n; m Fig. 118 and s; s and r; r and n. What are these angle pairs called? State the relations between m and r; s and n. The angle-pairs m and r, n and s are called opposite, т n S -D A or vertical, angles. Opposite angles are formed by two intersecting straight lines so that the sides of one angle lie in the same straight lines as the sides of the other, but in opposite directions from the vertex. From the measures of the angles (Fig. 118) it is seen that if two lines intersect, the opposite angles are equal. EXERCISES 1. Draw two intersecting straight lines and name the opposite angles. Express by equations the fact that the opposite angles are equal. 2. Two intersecting lines make one angle equal to 32o. Find the other angles. Clo dla 3. Find the sum of the angles just covering the plane surface around a point (Fig. 119). Express the result in the form of an equation. Fig. 119 64. Summary of angle relations. Express the relations for the following angles in the form of equations; in words; and by means of figures. 1. The angles of a triangle. 2. Two supplementary angles. 3. Two complementary angles. 4. Two opposite angles. 5. The acute angles of a right triangle. 6. The adjacent angles formed by two perpendicular lines. DRAWING ANGLES WITH THE PROTRACTOR 10 70 80 90 100 110 120 130 140 to 60 50 60 30 40 150 140 130 120 110 100 90 8'O 40 L091 ols A. 65. To draw an angle of a given size. To draw an angle of 45°, draw first a straight line, as ABC (Fig. 120). Then place c the center of B the protractor FIG. 120 at B and the zero-mark exactly on BC. Starting at the zero-mark, pass along the rim and place a point D at the 45° mark. Remove the protractor and draw a line from B passing through D. Angle CBD is the required angle. EXERCISES 1. Using the protractor and straightedge, draw an angle equal to 30°, 90°; 120°; 180°; 652°; 941 2. By means of the protractor draw a line perpendicular to a given line BC, at one of its points, as A. 3. Draw a triangle having a right angle. 66. To draw an angle equal to a given angle. In making designs like those of Fig. 121 one must be able to draw angles of the same size as the angles in the Fig. 121 required design. |