9. Is a rectangle necessarily a quadrilateral? Is it a parallelogram? A square? May a rectangle be a square? 10. Is a square necessarily a quadrilateral? Is it a parallelogram? A rectangle? A rhombus?

$107. Perimeters of Miscellaneous Figures.

1. Denote the perimeter of each of the forms in Fig. 1, p. 172, by p and write an equation (an expression of equality) like (I) in §105, showing the value of p for each figure.

2. Omitting the lines on which the forms join, write an equation showing the value of p when (2) and (6), Fig. 1, p. 172, are joined on y, which has the same value in both. 3. In the same way join (3) and (10) on a, and write an equation showing the value of p.

4. Join (2) and (7) in which x and y are equal. If the perimeter of the figure thus formed is 240 rods, find the value of x in feet.

5. What is the length of the perimeter of a figure like (6), Fig. 1, p. 172, whose sides are 2 in., a in., and b in. long? x in., 8 in., and z in. long? Like (10), whose sides are c ft., 2c ft., d ft., and x ft.?

Like (12), whose

sides are 2x rd., 4y rd., 2x rd., and 4y rd. long? 6. Write the perimeter (p) of each of the forms of Fig. 1, p. 175.

In such forms as those from (5) to (10), Fig. 1, p. 175, some line, or lines, must be found by subtracting others. In (6) for example, note that the ends are each x-y. The perimeter is then x+x+ (x−y)+y+y+ (x− y) −4x−y−y+y+y=4x-2y+2y=4x.

All sides not lettered must be expressed without using other letters than those given on the figure. The perimeter means the sum of all the lines that bound the strip, or surface, of the figure.

NOTE: One-half the difference of a line m and a line n is written

7. In (8), a=15', b=12', x=25' and y=30'. Find the length of the perimeter of the figure.

8. Find the area of (8) enclosed by the solid lines.

9. Make and solve other similar problems.

$108. Measuring Angles and Arcs.

Answer all you can orally.

We may measure the amount of turning of each clock hand in either of two ways:

(1) By the length of the circular arc passed over by the tip of the rotating hand; or,

(2) By the wedge-shaped space having its point at the hand-post, A, over which the stem (A-III), of the rotating hand moves.

FIGURE 2

While the clock hand turns from XII to III its tip moves over 1 quadrant of arc and its stem moves over a right angle.

1. While the hand turns from XII to VI over how many quadrants does its tip move? Over how many right angles does its stem turn?

2. Answer same questions for a turn of the hand from XII to IX; from XII to XII again; from XII around through XII to III.

3. Over what part of a right angle does the stem of the hand move while the hand is passing from XII to I? From XII to II? From III to IV? From VI to VIII?

4. Over what part of a quadrant does the tip of the hand pass in each case of problem 3?

5. Over what part of a right angle does the stem of the hand move while the tip moves 1 min. along the arc? In the same case over what part of a quadrant does the tip move?

Any wedge-shaped part of the face moved over by the stem of a clock hand as it turns around the hand-post is called an angle. The curve passed over by the tip of the hand, while the stem of the hand moves over the angle, is called the arc of the angle.

ILLUSTRATIONS.-The wedge-shaped spaces, having their points at A, and included between any two positions of the hand, as A-XII and A-I, A-I and A-III, A-I and A-VI, are all angles. The space swept over by the hand, A-I, as it moves around through II, III, IV, V, VI, etc., to IX is also an angle.

As the hand moves from A-XII to A-VI, that is, so that the two positions of the hand are in the same straight line, the hand moves over a straight angle.

6. A straight angle equals how many right angles?

7. The arc of a straight angle equals how many quadrants? How many quadrants make a complete

circle?

8. How many 5-min. spaces make the circumference of a complete circle? How many 1-min. spaces make the circumference of a complete circle?

9. What part of a right angle is passed over by the stem of the hand as its tip moves from one end to the other of a 1-min. space?

10. If lines were drawn from the hand-post to the ends of all the 1-min. spaces, the whole face of the clock would be divided up in

Instead of dividing the right angle up by radiating lines into 15 equal parts, the protractor divides the right angle into 90 equal parts, one of which is called the angular degree.

11. These same lines would divide the quadrant up into how many equal parts?

Each of these parts is a degree of arc.

12. How many angular degrees are there in a straight angle? How many degrees of arc in the arc of a straight angle?

13. How many angular degrees are swept over by a clock hand while moving entirely around once? How many degrees of arc in the circumference of a circle?

14. How many degrees of angle are passed over by the stem of the minute hand in two hours? In the same time

how many degrees of arc are passed over by the tip of the hand?

15. A sextant is of a circumference; how many degrees of arc are there in a sextant?

16. An octant is of a circumference: how many degrees of arc are there in an octant?

17. Study the protractors, Fig. 1, p. 177 and Fig. 1 below; notice how its marks are numbered. How many degrees of arc are there between the closest lines on the outer edge? How many degrees of angle are there between the lines which converge toward the center?

For smaller angles a shorter unit is the smaller unit is called the minute called the second, is a still smaller unit.

of a degree of angle and of angle. of the minute,

19. The arcs between the sides of the minute and the second of angle are the minute and the second of arc. How many minutes of arc in a quadrant? In a sextant? In an octant? In a circumference? How many seconds

of arc, in each case?