I. ON ANGLES 13 O Figure 13 Indeed, the sum of the intercepted arcs is a semicircle. Conversely, if two or more angles with the same vertex, and each adjacent to the next, (AOC, COD, DOE, EOA , Fig. 13) have a sum equal to two right angles, then their outer sides form a straight line. Indeed, these outer sides cut a circle centered at the common vertex of the angles at two diametrically opposite points, since the arc they intercept is a semi- circle. Remark. The angle AOA (Fig. 13) whose sides form a straight line is called a straight angle. 15b. Theorem. The bisectors of the four angles formed by two concurrent lines form two straight lines, perpendicular to each other. Let AA , BB (Fig. 14) be two lines which intersect at O and form the angles AOB, BOA , A OB , B OA, whose bisectors are Om, On, Om , On . We claim: 1◦. That Om, Om are collinear, as are On, On . 2◦. That the two lines thus formed are perpendicular. O Figure 14 First, Om is perpendicular to On because, since the sum of AOB and BOA is two right angles, half of each, mOB and BOn, must add up to a right angle. Applying the same reasoning to angles A OB and B OA, we see that Om and On are perpendicular. Therefore Om is the extension of Om, and likewise On is the extension of On. 16. An angle which is less than a right angle is called an acute angle an angle greater than a right angle is called an obtuse angle.

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