64. Kinds of angles. In Figure 20 the two lines AB and CD cutting each other at O form four angles, and the two lines are said to intersect.

Two angles such as AOC and COB, which have the same vertex and a common side OC between them, are called adjacent angles. In

In Figure 20, ZAOC is smaller than its adjacent angle COB. Let the line CD be rotated about the point O until ZAOC equals COB, as in Figure 21. Then the lines AB and CD are said to be perpendicular to each other and the angles formed are called right angles.

Definition. If one straight line meets another straight line making the adjacent angles equal, the lines are said to be perpendicular to each other and

the angles formed are called

draftsmen in drawing right angles and parallel lines. See

Figure 22.

Exercise 69

1. Draw an acute angle; an obtuse angle.

2. What kind of angle is made by the edges of the cover of this book? By the edges of the floor of your schoolroom?

3. Open your book until the edges form an acute angle; an obtuse angle.

4. Do the streets intersect at right angles in your city? Do you know of any streets that intersect at acute angles?

5. In Figure 19 read four pairs of adjacent angles and for each pair name the common side.

6. In Figure 20 is AOC larger or smaller than its adjacent angles?

7. In the adjoining figure are Za and b adjacent? Why? Are a and c adjacent? Why? Name a pair of adjacent angles in this figure.

cd

α

FIG. 23.

that one part of this The line along which

8. Take a piece of paper with a straight edge. Fold the paper so straight edge lies along the other part. the paper is folded makes what kind of angle with the straight edge of the paper?

9. In Figure 24 what kind of angle is there between the lines pointing (a) east and north; (b) east and southeast; (c) east and southwest; (d) east and northeast; (e) northeast and northwest?

10. What kind of angles are formed

by the edges of a cube?

NW

NE

SW

FIG. 24.

SE

11. In the figure of a pyramid on page 115, read an acute

angle; a right angle; two pairs of equal angles.

65. The sum and the difference of angles. If the angles AOB and DO'C are placed so as to be adjacent, then the angle AOC thus formed by their exterior sides is called the sum of angles AOB and DO'C. We may write ZAOB+2 DO'C= LAOC.

B

DO
FIG. 25.

A

FIG. 26.

If the DO'C in Figure 25 is placed on the ZAOB, so as to make Figure 27, so that O' falls on O, the side O'D falls along OA, and the side O'C falls within ZAOB, then the ZCOB thus formed is called the difference between ZAOB and DO'C. We may write

ZAOB-Z DO'C=COB.

In Figures 26 and 27 the angles are said to be added and subtracted graphically.

66. Complementary and supplementary angles. Two angles whose sum is a right angle are said to be complementary, and each is said to be the complement of the other.

In Figure 28, ZAOB and Z BOC are complements of each other.

Two angles whose sum is two right

B

FIG. 27.

A

B

A

FIG. 28.

angles are said to be supplementary, and each is said to be

the supplement of the other.

The unit usually used in measuring angles is of a right angle, which is called a degree. The symbol for degree is °. Thus we write 3 degrees as 3°.

This figure shows an angle of 1°.

FIG. 30.

Exercise 70

1. How many degrees in a right angle? In four right angles?

2. How many degrees in one-half of a right angle?

3. How many degrees in the angles between the hands of a clock at three o'clock? At one o'clock? At six o'clock?

At two o'clock? At four o'clock?

4. Through what angle does the minute hand of a clock rotate in 15 min.? In 30 min.? In 45 min.? In 60 min.? In 5 min.? In 10 min.? In 2 hr.? From twelve o'clock noon to twelve o'clock midnight?

5. What directions make angles of 45° with east? With south? With northeast?

6. Cut two angles from paper. Place them so as to show their sum and their difference.

7. Cut a triangle from paper. Mark the vertices A, B, and C. Cut off the corners and place them together so as to show the sum of the angles of the triangle.

8. Make a right angle by folding paper. Fold this angle so as to make an angle of 45°.

9. An engine on a turning table is headed east. In what direction is it headed after the table is turned to the right through an angle of 45°? 90°? 180°?

10. A wheel contains 16 spokes. What is the angle between two adjacent ones?

11. If ≤ A =65°, how many degrees in its complement? In its supplement?

12. How many degrees in 3 right angles? In 5 right angles? In a right angles? In x right angles?

13. How many right angles in an angle of 270°? Of 360°? Of do? Of m°?

14. Draw an acute angle AOB. Draw a line OC so that Z BOC is the complement of ZAOB. Draw a line OD so that BOD is the supplement of ZAOB.

15. Find the supplement of an angle of 40°; a°; x°; 5k°. 16. One angle contains X° and another contains Y°, and X+Y=90. Find X if Y has the following values: 40°; 50°; 75°; 1°; 90°.

17. If two angles A and B are supplementary, then A+ ZB=180°. Find B if A has the following values: 15°; 90°; 75°; 0°; 179°; 180°.

18. May two angles be complementary but not adjacent? 19. Draw two obtuse angles which are adjacent. Are they supplementary? Why?

20. Draw two acute angles which are adjacent. Are they supplementary? Why?

21. May the complement of an acute angle be obtuse? Why?

22. May the supplement of an acute angle be obtuse? Must it be so? Why?