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.