By carefully studying this chapter we shall gain a thorough understanding of angles. We shall measure and draw angles to learn about their size. We shall learn the names of various kinds of angles. We shall study the relations between angles in the same geometric figure, as in the triangle or in several lines intersecting in the same point (Fig. 73), and use these relations to solve some problems in angles.

We have said that an angle is a figure formed by two lines. It should not be inferred that two lines always form an angle. Point out lines in the class room which do not form an angle, no matter how far the lines may be extended. Such lines are known to us as parallel lines. The last part of this chapter takes up the study of angles formed by two parallel lines intersected by a third line (Fig. 74).

51. Size of an angle. In the clock (Fig. 75) the hands are shown to be together. Since the minute hand of a clock turns faster than the hour hand, a turn of the minute hand separates the two hands, as in Fig. 76. The hands then form an angle. When the hands of a clock move, the angle changes in size.

An angle may be formed by keeping point B (Fig. 77) fixed and then

FIG. 73

FIG. 74

C

turning a line from the position BC in the clockwise direction until it takes the position BA; or by turning a line from BA in the counterclockwise direction until it takes the position BC. The size of the angle depends entirely on the amount of turning necessary to carry the moving line from one side to the other, and not on the length of the sides.

FIG. 77

b

FIG. 78

A

Which of the two angles (Fig. 78) is the larger? Give a reason for your answer.

The curved arrow shown in the diagram is used to indi

cate the direction and amount of turning.

Two angles are equal if the same amount of rotation is needed to form them. If two angles are equal they can be made to coincide (fit).

Cut two angles from paper, and test them as to equality by placing one on the other. Tell which of the two is the larger angle.

52. Symbols used to denote angles. If we are to discuss angles, we must have a way of naming them. The symbol for the word angle is 4. For angles it is 4. There are various ways of naming an angle. We may use three letters (Fig. 79) one on each side and one at the vertex, and refer to the angle as ZABC, using the vertex letter as the middle letter.

a

B

FIG. 79

C

-A

A briefer notation is ZB, which really means 'an

angle whose vertex is B."

Sometimes a small letter, as a, is written within the angle, and the angle is then called a.

53. How angles are classified. Draw two lines in the position AB and AC (Figs. 84-88). Let a moving line turn about point A in the counterclockwise direction from the side AB to AC.

When the amount of rotation is equal to a quarter turn (Fig. 84) ≤BAC is called a right angle.

An angle less than a right angle is acute (sharp) (Fig. 85).

When AC makes a half turn (Fig. 86) a straight angle is formed. Thus, the sides of a straight angle are in the same straight line on opposite sides of the vertex.

Angles less than a straight angle and greater than a right angle are obtuse (blunt) (Fig. 87).

If the line AC makes a complete turn (Fig. 88) the angle is a perigon (round angle).

EXERCISES

1. State a time of the day when the hands of the clock form a right angle; a straight angle.

2. Point out obtuse angles in the classroom.

3. Open the arms of a blackboard compass so as to form an acute angle; a right angle; a straight angle.

4. Make a sketch of each of the following: right angle; acute angle; straight angle; obtuse angle; perigon.

A

b a FIG. 89

FIG. 90

B

5. Classify the angles a and b (Fig. 89).

6. Open your notebook, making two edges form a right angle; an obtuse angle; a straight angle.

7. Name an acute angle (Fig. 90); a straight angle; an obtuse angle; a right angle.

8. Classify the angles A, B, and C of triangle ABC (Fig. 90) and D.

9. Name the largest angle (Fig. 90); the smallest angle.

10. Write in symbols: angle B is less than angle D; angle BAC is greater than angle B; angle DCA is equal to angle CAD.

11. How many right angles are there in a straight angle? In a perigon? How many straight angles are there in a perigon?

12. Draw a triangle ABC (Fig. 91) making the base about 8 cm. long. Mark the angles x, y, and z, respectively. Cut the triangle from the paper and tear off the corners as shown in the diagram. Place them adjacent to each other as in Fig. 92. Show that the sum of the three angles of a triangle is a straight angle. This is an important fact of geometry.

x

y

FIGS. 91, 92

13. Draw a quadrilateral. Tear off the corners, and show as in Exercise 12 that the sum of the four angles is a perigon.

MEASURING ANGLES WITH THE PROTRACTOR

54. Protractor. The protractor (Fig. 93) is an instrument used mainly for measuring angles. The curved rim is divided into 180 equal parts, every tenth

rim, as OB, forms with the zero-lines, OX and OY, angles whose sizes may be read off on the outer and inner readings, respectively. Thus the measure of the straight angle XOY is 180, the measure of angle XOB is 30.