11. How many degrees are there in five-sixths of a right angle?

12. What part of a right angle is 18°?

13. What part is 60°?

14. What part is 72°?

3. Protractors. A protractor is an instrument which is used to find the size of an angle, or to construct an angle of any required size. Protractors are made of thin metal, ivory, cardboard, etc., and in various shapes of which the commonest are shown in the annexed pictures. They may be graduated to any degree of minuteness; but intervals of five degrees. are fine enough for our purpose.

If you have no protractor, you should make one out of card or stiff paper, copying one of the forms given here. The lower straight edge can be graduated so as to serve for a measuring ruler.

At the middle point of one edge, BA, of the protractor is a notch or dot, marked C in the diagram; this point is the vertex of any angle to

which the protractor is applied, and CA is placed directly on one side of the angle. The other side of the angle is indicated by the little lines at the edge of the protractor, having numbers which show the size of the angle in degrees. This second side is seldom drawn completely to the point C, because for convenience in use most protractors have an open space in the interior; but you will notice that if the lines around the rim were prolonged, they would all meet at the point C.

In the first picture of the protractor the angles are numbered from left to right; but in the second picture they are numbered from right to left according to the way in which angles are supposed to increase in size.

B

60

4. To measure an angle, as ACB, with the aid of a protractor, place the protractor as in the annexed figure, with the notch upon the vertex C, and the edge upon one side CA, so that the

point on the rim which indicates zero degrees may be on CA. Then observe the number of degrees marked on the rim of the protractor where it is crossed by the other side, CB, of the angle. This will be the number of degrees in the angle, if the protractor is graduated from right to left; but if from left to right, the number on the rim must be subtracted from 180° to show the size of the angle.

Estimate by the eye the size of the following angles, and then test your answers with a protractor.

5. To construct an angle of required size with the aid of a protractor. Suppose you wish to construct an angle of 130°. Draw a straight line CA of any convenient length. Place the protractor with its notch at C and the edge along CA. Then find on the rim of the protractor the mark which indicates the

angle 130°. Dot the paper at that point, remove the protractor, and draw the line CB through the dot. ACB will be the required angle.

B

130

C130

A

Construct the following angles with the aid of a protractor:

Construct angles equal to the following with the aid of a protractor:

Construct the following angles, ruling the lines but otherwise aided by the eye alone: then test your angles with a protractor: —

101. 40° and 140°, to have their vertices at the same point and one side in

common.

102. 130° and 50°.

103. 90° and 90°.

104. 60°, 90°, 120°, 90°, to have their vertices at the same point.

105. 45°, 135°, 80°, 100°.