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
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