10 I. ON ANGLES

11. We say that two angles are adjacent if they have the same vertex, a common

side, and so that they are located on opposite sides of this common side.

When angles AOB, BOC are adjacent (Fig. 8), angle AOC is called the sum

of the two angles.

The sum of two or more angles is independent of the order of the angles added.

To compare angles, we move them so they have a common vertex, and a com-

mon side, and lie on the same side of this common side. Assume that AOB, AOC

are placed in this manner. If, rotating around point O, we encounter the sides in

the order OA, OB, OC (Fig. 8), angle AOC is equal to the sum of AOB and

another angle BOC; in this case angle AOC is said to be larger than AOB, and

the latter is smaller than AOC; if, on the contrary (Fig. 9) the order is OA, OC,

OB, then angle AOC is smaller than AOB. The angle BOC which, added to the

smaller angle, yields the larger angle, is the diﬀerence of the two angles.

Finally, in the intermediate case in which OB coincides with OC, the two angles

are congruent (see 10).

In the interior of every angle BAC there is a ray AM which divides this angle

into two congruent parts; it is called the bisector of the angle. The rays contained

in the angle BAM make a smaller angle with AB than with AC; the opposite is

true for rays contained in the angle MAC.

An angle is said to be the double, triple, etc., of another if it is the sum of two,

three, etc., angles equal to this other angle. The smaller angle will then be called

a half, third, etc., of the larger one.

Remark. The size of an angle does not depend on the size of its sides, which

are rays (5), and which we must imagine as extending indeﬁnitely.

12. As we have said, an angle is determined by two rays, such as OA, OB

(Fig. 10). If we extend OA past point O to form OA , and in the same way extend

OB to form OB , we obtain a new angle A OB .

Two angles AOB, A OB such that the sides of one are extensions of the sides

of the other are called vertical angles.

Theorem. Two vertical angles are congruent.

Figure 10