The circumference of any circle is supposed to be divided into 360 equal parts, each part being called a degree.

If the arc AB contains 60 degrees, then the angle ACB at the centre is an angle of 60 degrees, expressed by 60°.

The lines AC, DE, through the centre, being perpendicular, each of the arcs AD, DC, CE, EA must contain 90°, and the angles ABD, DBC, . . . . are angles of 90°. A semicircle contains 180°, and the straight angle ABC contains. 180°.

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A triangle:

It has three sides and three angles.

A quadrangle :

It has four angles. Having four sides,

it is also called a quadrilateral.

A straight line joining two opposite corners of a quadrilateral is called a diagonal.

Figures contained by more than four straight lines are called polygons.

A straight line has evidently throughout its entire length the same direction.

Two straight lines which have the same direction are said to be parallel to one another.

Parallel straight lines cannot intersect. For if they did, at the point of intersection they Iwould have different directions, and would therefore have different directions throughout their entire lengths, and hence would not be parallel.

To construct with the protractor at the point A in the line AB an angle of any required magnitude, say 63° Place the centre of the protractor at A, and let the line

X

B

joining the centre with the point on the circumference which indicates 0°, rest along AB. At the point where the 63° line meets the circumference make a fine mark, C, on the paper. Removing the protractor, join AC. The angle BAC is of magnitude 63°.

Exercises.

All figures in this and succeeding exercises must be accurately constructed with instruments.

1. With the dividers (or compasses) take off on the ruler distances 8, 11, 17, 34 . . . . . . . . millimetres. With the points of the dividers mark on your paper points at these distances from each other. With the ruler draw straight lines joining each pair of points, thus getting straight lines of lengths 8, 11, 17, 34 . . . millimetres.

2. With the compasses describe circles having radii of lengths 5, 7, 10, sixteenths of an inch.

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3. With the protractor construct angles of magnitude 10°, 15°, 25°, 30°, 37°, 43°,

4. With the bevel construct a second set of angles of the foregoing magnitudes, using these angles to set the bevel.

5. Draw five straight lines of different lengths, and with the dividers and rule measure their lengths in inches and sixteenths of an inch. Measure also their lengths in millimetres.

6. Construct five angles, and, using the bevel, determine which is greatest and which least. Arrange them in order of magnitude. Using the protractor, measure their magnitude to the nearest degree.

7. Draw five straight lines of different lengths, and with the eye endeavor to judge their lengths (1) in inches and fractions of an inch, (2) in millimetres. Afterwards test the correctness of your judgment by actually measuring the lines.

8. Construct five angles of different magnitudes, and with the eye endeavor to judge the number of degrees in each. Afterwards test the correctness of your judgment by actually measuring the angles with the protractor.

9. With the eye endeavor to judge the lengths or heights of various objects in the room, at a distance from you. Afterwards test the correctness of your judgment by actually measuring the lengths or heights.

10. A and B being two distant objects and your eye being at C, endeavor with the eye to judge the angle which these objects subtend at your eye, i.e., the angle ACB. Afterwards sight the inside edges of the legs of the bevel towards A and B, and then placing the bevel on the protractor, roughly measure in this way the angle ACB, so correcting, if necessary, your judgment.

11. Draw any two lines of different lengths, and draw a line equal to their difference.

12. Draw any line, and draw another line three times as long as the former.

13. Construct two angles of different magnitudes, and with the bevel constructing two adjacent angles equal to them, form an angle equal to their difference. Measure with the protractor the number of degrees in the original angles and in the difference, and compare.

14. Construct two angles of different magnitudes, and with the bevel constructing two adjacent angles equal to them, form an angle equal to their sum. Measure with the protractor the number of degrees in the original angles and in the sum, and compare.

15. Construct an angle of 30°. With the bevel construct two other angles equal to it, one on each side of the first, the three bounding lines radiating from the same point. What positions do the outside lines of your figure occupy with respect to each other, and why? Test with an instrument.

16. Construct an angle of 60°. With the bevel construct five other angles equal to it, each adjacent to the preceding, the bounding lines all radiating from the same point. What positions do the first and last lines of these angles occupy with respect to each other, and why?

17. In the figure of the preceding question, if O be the point from which the lines radiate, measure off with the dividers on these lines equal lengths, OA, OB, OC, OD, OE, OF. What do you observe as to the lengths AB, BC, CD, DE, EF, FA?

18. Fold a piece of paper so as to get a straight crease. Fold the crease over on itself. How many degrees in each of the four angles so obtained, and why?

19. With a needle mark two points. Join them, using ruler and a fine pencil. Turn the ruler over to the other side of the two points and again join them. What quality in the ruler may you test in this way?

20. At points on your paper some distance from one another, construct two angles, as nearly as you can judge, equal. Test with an instrument the correctness of your judgment.

21. Through what angle does the minute-hand of a clock move in 20 minutes? Through what angle does the hour-hand move in the same time?

22. Describe a circle, and, supposing it intended for the face of a clock, mark the points where the usual Roman numerals should be placed.

23. One side of a piece of paper being a straight line, tear the remaining boundary into any irregular shape. With your protractor convert this paper into a protractor, so as to mark angles at intervals of 10°, the markings being on the irregular edge of the paper.

Construction of Triangles.

1. Take a line AB of any length. First with A as centre, then with B as centre, and in both cases with the same radius AB, describe portions of circles so that they intersect, as indicated, at C. Then the three lines AB, BC, CA are all equal. The tri

angle CAB, which has thus all its sides equal, is called an equilateral triangle.

Adjust the bevel to each of the angles of this triangle, and compare their magnitudes.

Construct equilateral triangles whose sides are 14, 21, 30, 40. . . sixteenths of an inch.

Apply the bevel to all the angles of these triangles, and compare their magnitudes.

Cut accurately any one of these equilateral triangles from the paper, and, clipping off the angles, fit them on one another, and on the angles of the other equilateral triangles, so as to compare their magnitudes.

The result of our observations is that the angles in an equilateral triangle are equal to one another, and are equal to the angles in any other equilateral triangle.

Using the bevel, construct three angles adjacent to one another, in the way indicated in the annexed figure, each angle being equal to the angle of an equilateral tri.

NOTE.-It is well to mark on lines and angles their magnitudes, when known.