THE ANGLE

19. Definition and Notation. An angle is a figure formed by two rays having a common origin. The two rays, AB and AC, are called the sides, and A, their origin, is called the vertex, of the angle.

B

The outstretched fingers, the hands of a clock, the spokes of a wheel, the divisions of the compass, the gable of a roof, and many similar illustrations readily suggest the frequent occurrence and great importance of angles.

A

C

The methods of naming angles are shown in the figures below.

A single angle at a point may be designated by a capital letter placed at its vertex, as angle O.

When several angles have the same vertex, each angle may be designated by three letters; namely, by one letter on each of its sides, together with one at its vertex. The letter at the vertex is read between the other two, as

angle DAC, angle CAB, etc.

Sometimes an angle is denoted by a small letter placed between its sides near the vertex, as angle m.

b

The sides may be denoted by small letters. The angle ab is the angle formed by the rays a and b. This notation is, however, used less frequently than the preceding.

The word "angle" is frequently replaced by the symbol Z. The symbol is used as an abbreviation for "angles."

EXERCISES

1. How many angles may be found in the letters N and W? 2. From a given point draw three rays; four rays; five rays. How many angles are formed in each case?

3. Through a given point draw two lines; three lines; four lines. How many angles are formed in each case?

4. Draw two angles, making them as nearly equal as you can. Test their equality by copying one of the angles on tracing paper, or by cutting out one of the angles, and applying it to the other.

C

D

5. The figure shows how a strip of paper having a straight edge may be used to copy an angle. Place the straight edge on the side AB of ∠BAC, and mark on it the position of the vertex A. Mark also the point D, where the side AC intersects the other edge of

the paper. Then ∠DAE=∠CAB.

E

B

A

With the aid of this paper strip draw an angle equal to ∠CAВ.

6. If you extend the sides of ∠CAB beyond C and B, do you change the size of the angle?

7. Draw a ray and place your pencil on it. Revolve the pencil on the flat surface of the paper, using one extremity of the pencil as a pivot. Observe that each position of the pencil indicates a different angle. The pencil will eventually return to its first position. This rotation evidently gives a picture of every possible angle.

α

b

a

b

8. Show that in the figures a line could revolve from the position a to the posi

tion b, or from b to a.

This fact is indicated by the arrowheads.

CLASSIFICATION OF ANGLES

(A) THE THREE FUNDAMENTAL ANGLES

20. From the preceding exercises may be derived a general idea of the magnitude of an angle. It appears that the magnitude of an angle depends on the amount of revolution necessary to turn a line through the angle about the vertex as a pivot. The revolving line is said to generate or describe the angle. The size of an angle is independent of the length of its sides.

21. A round angle, or perigon, is an angle whose magnitude is indicated by a complete revolution of the generating line. 22. All round angles are equal.

23. A straight angle is an angle whose sides are in the same straight line, on opposite sides of the vertex; as ∠AOB.

A

24. All straight angles are equal. Why? 25. Two angles having the same vertex and a common side between them are called adjacent angles; as the AOB and BOC.

EXERCISES

A

0

B

C

1. Can two angles have a common side and a common vertex without being adjacent?

2. In the annexed figure ∠AOC is a straight angle. If a ray OB revolves from the position OA toward the position OC, two angles are formed, ∠m and Zn. At

C

Y

n/m

B

A

first ∠m is less than ∠n. Finally, however, ∠m will be greater than ∠n. Hence one position of OB must have made the angles equal. Indicate approximately this position of Oв in the figure.

3. Take a piece of paper having a straight edge AB and fold it so as to bring the point B on the point A. Open the paper and name the crease OC. Then ∠AOC = ∠BOC.

26. A right angle is an angle such that the adjacent angle formed by extending one of its sides beyond the vertex is equal to it. Thus, in the figure, ∠BOC is a right angle. 27. A right angle is half of a straight angle. Why? 28. All right angles are equal. Why?

29. The sides of a right angle are said to be perpendicular to each other; thus OC in the figure is perpendicular to OB, and OB is perpendicular to OC. This is sometimes abbreviated as follows: OC⊥ OB,OB ⊥ OC. If OC is perpendicular to AB, O is called the foot of the perpendicular Oс.

30. Round angles, straight angles, and right angles are so important that they are taken as standards with which other angles are compared.

31. If three rays, a, b, and c, are drawn from the same point, as shown in the accompanying figure, the following relations exist:

1. Lac is the sum of ∠ ab and ∠bc.

2. Lab is the difference between Lac

and bc.

3. Zac is greater than either Zab or ∠bc.

4. If ∠ab=∠be, the ray b is the bisector

of Lac.

C

32. Two angles A and B must be in one of the following three relations to each other: ∠A> ∠B, ∠A = ∠B, or ∠A <∠B.

(B) THE THREE OBLIQUE ANGLES

33. An acute angle is an angle less than a right angle, as the angle COD.

34. An obtuse angle is an angle greater than a right angle but less than a straight angle, as angle AОВ.

35. A reflex angle is an angle greater than a straight angle but less than a round angle, as angle MON.

36. Acute, obtuse, and reflex angles are sometimes called oblique angles, and intersecting lines not mutually perpendicular are said to be oblique to each other.

EXERCISES

1. What kind of angle is each of the following angles of

4. Point out right angles in the construction and equipment of the schoolroom; in the exterior of a house; in the street,