Ex. 1.

What is the path of a moving point?

Ex. 2. What geometric figure is, in general, generated by a moving line? by a moving surface?

Ex. 3. Can a straight line move so that its path is not a surface?

Ex. 4. How does a stone cutter use the straight edge to determine whether a surface is plane?

Ex. 5. What kind of surface is represented by each wall of a room? Ex. 6. What kind of surface is represented by a gas-pipe?

ANGLES

22. If a straight line 04 revolves about one of its points O until the line reaches the position OB, then the amount of this rotation is called "the angle AOB." Obviously the amount of rotation, and hence the angle, does not depend upon the length of the line which rotates.

The lines OA and OB are called the sides and

the point o the vertex of the angle AOB.

منے

The student should note that the preceding statement is not a definition, but merely an explanation of the term angle. No definition of this term exists that is free from objections.*

23. Notation. If three letters are used to denote an angle, the vertex letter should be read between the others; as angle ABC, angle EOF. A single letter at the vertex denotes the

*The definition that is at present most widely used is the following one: "An angle is the figure formed by two straight lines diverging from a point." This definition, however, not only fails to explain which part of the figure really constitutes the angle, but it also makes use of the undefined term 'diverge.' Moreover it is not applicable to angles greater than 180°.

largest angle at this vertex (if there be several at this point). Thus, angle DOF may be read "angle o," angle ABC may be read "angle B."

Frequently an angle is also designated by a number, or italic letter, placed within it, as angle 1, angle 2, angle m.

Often a curve is drawn to point out more clearly which angle is meant ; as angle 2, and angle 3. An arc placed close to a number shows which angle is designated. Thus, angle MXP may be read "angle 3," and

angle NXQ may be read "angle 4."

24. To bisect an angle means to divide it into two equal parts.

Thus, BD bisects angle ABC, if angle

angle DBC. BD is called the bisector

ABD
of angle B.

25. DEF. A straight angle is an angle whose sides lie in the same straight line

but extend in opposite directions, as ABC. C

B

A

26. DEF. A right angle is an angle equal to one half of a straight angle.

Thus, if OC bisects the straight angle AOB, angle 3 and angle 4 are right angles.

27. DEF. An acute angle is an angle less than a right angle; as angle 5.

28. DEF. An obtuse angle is an angle greater than a right angle, but less than a straight angle; as angle MNO.

29. DEF. A reflex angle is an angle greater than a straight angle, but less than two straight angles; as angle 6.

An angle denoted by the usual methods does not signify a reflex angle, unless designated as 'reflex angle' or indicated by an arc.

30. DEF. Acute, obtuse, and reflex angles are called oblique angles.

31. DEF. Two lines are perpendicular to each other if they meet at right angles; as AC and BO.

The point of meeting (0) is the foot of the perpendicular.

32. An angle is measured by finding how many times it contains a certain unit. The usual unit is the degree, or one-ninetieth (%) of a right angle. A degree is divided into sixty equal parts called minutes, and a minute into sixty equal parts called seconds. Degrees, minutes, and seconds are expressed by symbols, as 6° 50′ 12′′. Read six degrees, fifty minutes, and twelve seconds.

33. DEF. Adjacent angles are two angles that have a common vertex, and a common side between them; as angles 4AOB and BOC.

34. DEF.

A

Two angles are vertical angles if the sides of each

are prolongations of the sides of the other; as angles 7 and 8, or angles 9 and 10.

35. DEF. Two angles are complementary if their sum equals a right angle.

Each is then called the complement

of the other. Angles 5 and 6, or MPN and NPO, are complementary.

P

M

36. DEF. Two angles are supplementary if their sum equals

a straight angle.

Each angle is then called the supplement of the other. Angles 1 and 2, or angles 3 and 4, are supplementary.

Ex. 7. How many degrees are in a right angle? In a straight angle? In one half a right angle?

Ex. 8.

What is the angle made by the two hands of a clock at
At six o'clock ? At two o'clock? At five o'clock ?

three o'clock?

Ex. 9.

What is the angle made by the hands of a clock at 1 p.m. ? At 2:30 P.M.? At 5:30 P.M.?

Ex. 10. Over an angle of how many degrees does a spoke of a wheel sweep when the wheel makes of a revolution? of a revolution? 2 revolutions?

Ex. 11. How large is each angle at the center if a pie is divided into 5 equal parts? 6 equal parts?

Ex. 12. What angle is formed by lines drawn towards north and northeast? Towards S. and S.E.? Towards N.W. and S.W.?

Ex. 13. Over what angle does the large hand of a watch sweep in 10 min. ? 15 min. ? 30 min. ? 45 min. ? 1 hr. ?

Ex. 14. In the diagram of Ex. 15 read by three letters: a, b, Ld, < (a + b).

Ex. 15. In diagrams similar to the one given here find the numerical values of the required angles :

(a) If a = 30°, and ≤ b = 40°, find ▲ AOC.

(b) If Zb

(c) If Zb

= 35°, and c = 10°, find / BOD.

40°, c = 10°, and d

50°, find / BOE. (d) If ▲ AOC = 60°, and ≤ b = 40°, find a. (e) If ▲ AOD = 90°, ▲ a = 35°, and ≤ c = 10°, find Zb. (f) If ZAOE = 110°, a = 20°, and ≤ d = 30°, L BOD.

(g) If ▲ AOC = 60°, and ≤ a = 2b, find

a.

(h) If ▲ AOD= 75°, and ▲ a = 2b = 2c, find c.

E

find

Ex. 16. In the preceding diagram, which angles are adjacent to ZBOC? to Z COD? to ▲ BOD?

Ex. 17. In diagrams similar to the one shown, if ≤ 0 = 90°: (a) Which angle is the complement of a?

(b) Which angle is the complement of ZAOC? (c) Which angle is the complement of BOE? (d) If ≤ d = 20°, find

AOD.
COE

=

55°, find a.

(ƒ) If ▲ AOC = 55°, and ≤ d = 15°, find c.

(g) If Za = 2 b = 2 c = 2d, find a.

E

D

B

a

Ex. 18. How many degrees are in the complement of 30° ? Of 35° ? Of right angles? Of n° ? Of1 of a right angle ? Of (10 + x)° ?

n

Ex. 19. How many degrees are there in an angle that is twice its complement?

Ex. 20. In diagrams similar to the annexed one, if FBA is a straight line,

*(h) If / FBC = 140°, and Z ABD = 80°, find ≤n.

*

(i) If ▲ ABD = 80°, ≤ n = 35°, and ≤ CBE = 85°, find ≤p.

Ex. 21. How many degrees are in the supplement of 20° ? of 140° ?

of straight angles? of n degrees? of (50 - 3 x)° ?

* Exercises denoted by (*) are difficult.