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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