I. ON ANGLES 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-
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 .
That the two lines thus formed are perpendicular.
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.