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coincide, then the angle from the initial side of the first to the

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39. Placing Angles on Rectangular Axes. To place any given angle on a pair of rectangular axes in the plane of the angle, put the vertex at the origin and the initial side on the x-axis extending to the right; the terminal side will then fall in one of the four quadrants (or, if the angle is a multiple of a right angle, on one of the axes). If the terminal side falls in the first quadrant, the angle is said to be an angle in the first quadrant, etc. In Fig. 50, a is a positive angle


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FIG. 50





in the first quadrant, B is a negative angle in the fourth quadrant, & is a positive angle in the fourth quadrant.

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1. What angle will the minute hand of a clock generate in 2 hr. 24 min. 10 sec. ? 5250


2. A flywheel is running steadily at the rate of 450 revolutions per

What angle does one of its spokes generate in 2 sec.? In 1.2 sec.? 3/50

3. By means of a ruler and a protractor, construct the following angles and their sums; check by adding their numerical measures.

(a) - 75° and 125°.

(d) - 60° and — 36°.

(b) 66° and 30°.

(e) 485° and 55°.

(c) 45° and 30°, and 70°. (f) - 750° and 30°.

4. With some two of the angles just given verify a + ß = ß + α.

5. (a) Construct 27° + 85° + (− 45°)+ 135°.

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6. If a wheel is rotating 120° per sec., how many revolutions does it

make per minute ? how many per hour? How many degrees does it turn

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7. Express an angular speed of 2.5 revolutions per second in degrees per second; in revolutions per minute; in degrees per minute.

8. A flywheel rotates at the rate of 40 revolutions per minute.

Through what angle does one of its spokes turn in a second?

9. Reduce an angular speed of 3.4 revolutions per second to degrees per second; to degrees per minute; to revolutions per minute.

10. Find the angular speed of the rotation of the earth on its axis (a) in revolutions per minute; (b) in degrees per second.

11. Construct a right triangle whose sides are 3 and 4; construct an angle which is 3 times the smaller angle of this triangle.

12. Construct a right triangle with hypotenuse = 12 and one side 6; construct an angle equal to one fourth of the larger acute angle of this triangle.

13. Construct an angle 3.5 times the smallest angle of the triangle in Ex. 12.

14. Every acute angle is a positive angle in the first quadrant; construct and place on the axes a positive angle in the first quadrant that is

not acute.

15. Every obtuse angle is a positive angle in the second quadrant; construct and place on the axes a positive angle in the second quadrant that is not obtuse.

16. Construct the following angles and place them on the axes: (a) — 150°; (b) 285°; (c) 480°; (d) 570°; (e) — 225°; (ƒ) 450°. 17. In what quadrant is each of the following angles: 459°, 682°, 725°, 100°, – 1090°, ± 85°, ± 95°, ± 175°, ± 185°, ± 265°, ± 275°, +355° ?

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40. Congruent Angles. If the difference of two angles, a and B, is n times 360° (where n is one of the numbers 0, 1, 2, 3, etc.), they are said to be congruent angles and we write aß; read: a is congruent to B. Thus 15°375°, - 172°


188°, etc. If two angles y and & are not congruent, they are said to be incongruent, and we write y8, 87. Thus, 45° 400°. To prove that two angles are congruent it is necessary and sufficient to show that their difference is either 0 or a multiple of 360°; that is, if aẞ, a − ß = ± n ⋅ 360°; and conversely.

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41. Properties of Congruent Angles. If two congruent angles are placed on the same pair of axes, their terminal sides will coincide. For example, any two of the angles 50°, 410°, - 310° are congruent; when placed on the same axes their terminal sides all coincide. If two incongruent angles bẹ placed on the same axes, their terminal sides will not coincide.* This is the geometric equivalent of § 40.

(1) Every angle obtained by putting n = 0, 1, 2, 3, etc. in the formula a±n 360° is congruent to a; conversely every angle congruent to a is found in this set. (Use § 40.)

(2) If a is any angle whatever, there is one and only one angle between 0° and 360° (0° included, 360° excluded) which is congruent to α. For if a is an angle of any size, the addition to a and subtraction from a of successive multiples of 360° (360°, 720°, 1080°, etc.) will give all angles congruent to a, and obviously one and only one of these lies between 0° and 360°.

(3) If ay and if ẞy, then a B. That is, if each of two angles is congruent to a third angle they are congruent to each other. Proof: α- = ±m • 360°, γ whence, adding: a−ẞ=(±m±n) 360°, that is, aß.


= ±n

γ B=

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*The word congruent is thus equivalent to the word superposable as used in geometry; but we must remember that two angles are superposable if and only if it is possible to make them coincide vertex with vertex, initial side with initial side, terminal side with terminal side. That such angles are not identical is evident in such practical instances as rotating machinery; the motion of a flywheel, 30° per second, differs essentially from that of a wheel turning 410° (or from that of one turning - 310° per second). See § 37, p. 51.

(4) If the same angle be added to (or subtracted from) each of two congruent angles, the results will be congruent angles.

Given aẞ, to prove: (a) a +y=B + y, (b) α- y = B―y. (a) (a+y)-(B+y)= α − B = ±n 360°; hence a+y≈ß+y (b) (α-y)-(B − y) = α − ß = ±n 360°; hence a yß-y.

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(5) The negatives of congruent angles are congruent.

Given a ẞ, to prove that (— α)~(—ẞ).

Proof: (-)-(—α)=α- ẞ= ±n · 360°; hence (—ß)~(—α). By (4) the transposition of a term from one side of a congruence to the other with change of sign is permissible; e.g. from 45° – 350° ≈ 55° follows 45° 55° + 350°; from a + 150°≈ B + 180° follows α - ẞ 30°. By (5) it is permissible to change the signs of all terms of a congru- 18° 3 x 63° follows ence; e.g. from 2 x x45° and x 45°. It is not ordinarily permissible to multiply or divide both sides of a congruence by any number (except to multiply by an integer); e.g. from 30° 390° it does not follow that 10° ~ 130°.

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1. Draw figures on polar coördinate paper to illustrate (4), § 41, when (a) α = 30°, B = 390, y = 20°; (b) α = 90°, B = 270°, y = 45°; (c) α = 72°, B = 432°, y = 72°.

β ==

2. Taking α = 60°, - 300°, y =— 50°, 8310°, draw a figure showing that (a) α ≈ ß; (b) y ≈d; (c) α — y≈ ß — d.

3. Find the angle between 0° and 360° which is congruent to each of the following: (a) — 42° 13′; (b) — 842°; (c) 364° 23'; (d) 360°;


- 90°; (ƒ) 420°; (g) 2700°.

4. Solve for x the congruence 27°. x360° + 2 x.

Ans. x=9°n 120°.

5. Find 3 values for x which satisfy the congruence 3x-70°~150°—x. Ans. x- 35°, 55°, 145°.


6. Find the smallest positive value of x which satisfies the congruence : x + 200° 40° - 3x. x = 50°. 7. Prove that the sum of the interior angles of a convex polygon is congruent to 0° or 180° according as the number of sides is even or odd. 8. Compare a rotational speed of 30° per sec. with a speed of 390° per sec.

9. Reduce an angular speed of 390° per second to revolutions per second; to revolutions per minute.

10. If a ẞ and y≈d, prove that a + yẞ+d, and a

γ α β - δ.


42. Measurement of Angles. An angle may be named and used before it is expressed in any system of measurement. Thus, we may refer to an angle A of a right triangle whose perpendicular sides are 16 in., and 24 in., respectively; and we can compute tan A= 24/16=1.5, etc., without measuring A in terms of any unit angle. General theorems like the law of sines remain true in any system of measurement.

The measure of an angle - say 36°— consists of two distinct ideas the unit angle (in this example, one degree) and the abstract number (here 36) which expresses the numerical measure of the angle in terms of the chosen unit. The elementary units are defined in § 37. For many purposes it is convenient to use another unit angle called the radian.

43. Radian Measure of Angles.* A radian is a positive angle such that when its vertex is placed at the center of a circle, the intercepted arc is equal in length to the radius.

This unit is thus a little less than one of the angles of an equilateral triangle; in fact it follows from the geometry of the circle, since the length of a semicircum ference is πr, that


(1) π radians = 180°, where T = 3.14159, whence 1 radian = = 57° 17′ 44′′.806, or 57°.3 approximately. Inversely 1° = .01745 radians. It is easy to change from degrees to radians and vice versa by means of relation (1), which should be remembered. Conversion tables for this purpose are printed in Table IV.

FIG. 51

44. Use of Radian Measure. It is shown in geometry that two angles at the center of a circle are to each other as their intercepted arcs; therefore if an angle at the center is measured in radians and if the radius and the intercepted arc are measured in terms of the same linear unit, their numerical measures satisfy the simple relation:

arc= angle radius.

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