12. What fractions of a right angle are the angles formed by the hands of a clock, from hour to hour, between 12 and 6 o'clock? What fractions of a straight angle? How many degrees in each angle? State in each case whether the angle is acute, right, or obtuse.

13. How many degrees in the angles formed by the hands of a clock when the time is 3.12, 11.48, 10.48, 8.24, 9.36, 6.20, 7.22 (railroad notation)? (Remember that the minute hand travels 12 times as fast as the hour hand, and that a minute division on the face of the clock represents 6°.)

14. In how many minutes does the minute hand revolve through an angle of 90°? 45°? 270°? In how many hours does the hour hand describe these angles?

15. What angle is described in an hour by the minute hand? by the hour hand?

16. A man, setting his watch, moves the minute hand forward half an hour and then moves it back 8 min. How many degrees in the angle between the first and the final position of the minute hand?

17. The driving wheel of an engine revolves 10 (20, 30, x) * times per minute. In what time does one of the spokes turn through a right angle?

18. A screw required 10 complete turns before it was firm in the wood. The depth of the hole was found to be in. How far did a turn of a straight angle drive it?

19. What is the complement of 24° 17′? of 79° 11′? of 46° 34′ 10′′? 20. Find the sum of the following angles: 26° 47′3′′, 44°22′32′′, 68° 51′ 48′′, 39° 58′ 37′′.

21. Find the supplements of the following angles: 163°17′, 48° 34′, 94° 52′ 21′′.

22. The supplement of an angle is 8 times its complement. Find the value of the angle in degrees, minutes, and seconds.

23. Reduce to degrees, minutes, and seconds of a straight angle.

24. The accompanying figure shows the card of the mariner's compass. Count the number of equal parts into which the "points of the compass" divide the round angle. How many degrees in each of these parts? Draw a compass card.

26. Determine the course of a ship, that is, the point of the com

pass toward which a ship is sailing, in each of the following cases:

EXERCISES

REVIEW AND EXTENSION

1. From a point O as a common origin draw six rays indicated in order by the letters A, B, C, D, E, F. Express ∠AOD and ∠ BOF

each as the sum of three angles; ∠AOC

and ∠BOD each as the difference of

two angles; ∠AOB + ∠BOD-∠COD as a single angle.

2. If ∠AOB = ∠BOC, what name is given to OB?

3. In the above figure,

F

D

E

C

B

A

0

if ∠AOB = ∠COD, show that ∠AOC = ∠BOD;
if ZAOC = ∠BOD, show that ∠AOB = ∠COD;
if ∠AOB = ∠COD, while ∠AOD = 3 ∠AOB and
Z BOE = 3 ∠COD, show that ∠AOD = ∠B0E;

if ∠AOB = ∠AOC, and ∠COD = + ∠COE, while ∠AOC = ∠COE, show that ∠AOB = ∠COD. Express in words the principles underlying these relations. 4. On one side of a straight line, and having a common vertex, are situated (a) three (four, five) equal angles; (b) four angles, of which each after the first is twice as large as the preceding one. How many degrees in each angle?

Let x = number of degrees in the first angle.

5. There are four angles about a point, of which each after the first is three times as large as the preceding angle. How many degrees in each angle?

6. What is the result in Ex. 5, if the second angle is three times the first, and the third and fourth are each equal to the sum of the two preceding angles?

7. One of two supplementary angles is 2 (3, 4) times as large as the other. How many degrees in each angle?

8. There are five angles about a point. Four of them have the following magnitudes: 74°, 101°, 59°, 94°. How many degrees in the fifth angle?

9. Given Za = 40°, ∠ b = 30°, ∠ c = 20°, ∠d = 10°. Draw angles

equal to 2a + ∠b, ∠b + ∠d, Z b − Z c, Z b + Za - Zd.

Za+Zb

10. Given ∠a = 150°, ∠ b = 60°. Draw angles equal to

Zb 2

3

3

4

La-2b; La+La+20.

11. How many degrees does the minute hand of a clock traverse in 1 hr.? in (1) hr.? in 30 (20, 10, 25) min. of time? in 5 (12, 24, 36) min. of time?

12. In what time does the minute hand of a clock describe an angle of 45°? 90°? 180°? 270°?

13. Change to the lowest indicated denominations 20°24′; 30°30′; 179° 59′ 60".

14. Draw an angle of 50° and bisectit

with the aid of the protractor.

15. In the figure m and n are the bisectors

of the two supplementary-adjacent angles.

Prove that m 1 n.

m

n

16. If an angle is bisected, prove that the supplements of the two equal angles are equal.

17. What kind of angle is formed by the

bisectors of two vertical angles?

18. In the figure m In, and Za = ∠α'. Show that Zb = Zc.

19. The sum of an acute angle and an ob

tuse angle cannot exceed how many degrees?

m

αα'

b

C

20. If ∠a is greater than ∠b, show that the supplement of Za must be less than the supplement of Zb.

21. How many degrees in an angle which is 12o less than its supplement? 18° greater than its complement?

22. Find ∠A and B, if (∠ A + ∠B) = 48° 16′20′′, while (∠A - ∠B) = 22° 52′ 17′′.

23. Find the value of one half the supplement of 65° 11' 31". 24. How many complete revolutions in an angular magnitude of 7832°? How many degrees between the initial and final positions of the revolving ray?

TRIANGLES

45. Experience teaches us that two straight lines do not inclose a space. If, however, the sides of an angle are crossed by a straight line, we obtain a closed

space.

A triangle is a figure bounded by three straight lines. The points of intersection of the lines are called the vertices,

and the segments joining the vertices are called the sides of the triangle.

The vertices are denoted by capital letters and the sides by small letters. Sometimes it is convenient to use corresponding letters, that is, the side opposite ∠A is denoted by the corresponding small letter a.

The sum of the three sides of a triangle is called the perimeter.

46. The Angles of a Triangle. Exterior Angle. The following exercises are intended to develop a clearer insight into the mutual relations of the angles of a triangle.

EXERCISES

1. Point out triangles in the exterior of a house; in the street; on the surfaces of certain solids (e.g. pyramids).

2. Draw five triangles of different shapes, and measure the sides of each. Tabulate your results. Can any three segments be used for the sides of a triangle ?

3. Measure the angles of the triangles and tabulate. You will find that the sum of the three angles in each case is equal