AOB+BOD + DOC + COA = k 360°,

AOCCOB + BOD + DOA = k 360°.

where k, as in § 18, denotes any integer, positive, negative, or null.

In like manner, if a, ß, y, d, are any lines in one plane, —

§ 20. Before proceeding further we must lay down rules concerning the directions of certain lines which we shall have to treat of in their relations to angles.

a. When speaking of an angle, we shall always regard the vertex as a point of reference or origin with respect to this angle; and we will define the directions of the initial and terminal lines (§ 12) to be always positive, when considered relatively to the angle. Thus, when we speak of the angle between two lines, we always mean the angle between their positive directions.

In each of the figures 16, 17, 18, and 19, for example, o, the vertex of the angle q, will be an origin, and the directions of the initial and terminal lines ox and Oв will be positive.

b. Through o, making with the initial line ox an angle of 90°, draw or. The lines xx' and YY' are called axes. We define ox and or as the positive directions of the axes; ox' and or are therefore their negative directions. These are also the positive and negative directions of all lines parallel to the axes, when considered relatively to them. Thus PM and P'N are positive, while PM' and P'N' are negative; and LM and L'M' are positive, while LN and L'N' are negative. Relatively to the angle XOB, OB is positive, and OB' negative; but relatively to the angle XOB', OB is negative, and oв' positive.

c. Whenever in the same problem we have to treat of two or more angles whose vertices and initial lines do not coincide, we can either refer all the angles to the same origin and to the same initial line, or we can regard each vertex as an origin, and each initial and terminal line

as positive, with respect to the angle to which it belongs. If the latter method is adopted, care must be taken not to confound a direction which is positive with respect to one angle with a direction which is positive with respect to a different angle; for these directions may be, when compared with one another, directly opposite.

d. When we have q= 180°, or q = 270°, the terminal line (OB) will have the negative direction of the axis: we shall, however, still consider Oв to be positive, when regarded as the terminal line of q. Suppose, for example,

we have the right triangle BOC (Fig. 20). The base oc is negative; and the hypothenuse Oв and perpendicular CB are positive. Now, suppose to increase until Oв coincides with oc. We shall now regard our right triangle to be one whose perpendicular is zero, whose base oc is negative, and whose

hypothenuse oв is, as before, positive.

$ 21. The following principles of Plane Geometry must be borne in mind:

a. The sum of the angles of a plane triangle is equal to 180°.

b. The sum of the two acute angles of a plane right triangle is equal to 90°. Either of these angles is then the complement (§ 5) of the other.

c. When two plane right triangles have an acute angle and a side of the one equal to an acute angle and the homologous side of the other, the triangles are equal, and all of their homologous parts are equal.

d. When two plane right triangles have an acute angle of the one equal to an acute angle of the other, the triangles are similar, and their homologous sides are therefore proportional.

e. The square on the hypothenuse of a right triangle is equal to the sum of the squares on the other two sides. f. In any triangle the greater side is opposite the greater angle, and the converse.

g. The sum of two sides of a triangle is greater than the third side.

FIG. 21.

B

h. Two angles are equal when their sides are respectively perpendicular; but we must be careful to take the sides of the respective angles in the same order, and to measure the angles in the same direction. (v. § 14.) In Fig. 21, for example, the angle B'AC-angle XOA, for both angles are measured in the positive (§ 14) direction, and B'AC is formed by AB', which is perpendicular to ox, and AC, which is perpendicular to OA. Similarly, angle

BAC' =XOA, and angle X'AB' = ABO.

B

i. The chord of an arc of 60° is equal to the radius.

EXAMPLES.

1. To what quadrant does each of the following angles belong? 289°, 368°, 510°, 640°, 1178°, 80°, 188°, 1722°.

2. Represent by figures the following angles, where in each case = 60°, and 0 = 30°

360°,

- 270°, ¢ + 0 + 60°.

90°, p

3. Change to circular measure each of the following angles: 88° 2′, 271° 53', -18° 7′, 60°, 390°.

4. The circular measures of certain angles are as fol

5. Find the lengths of the following arcs in a circle whose radius is unity: 60°, 30°, 150°, 330°, 18°, 268°, 135°. 6. Find the lengths of the above arcs in a circle whose radius is 6.

7. Taking the earth's equatorial radius to be 3963 miles, find the length of an arc of 1° on the equator.