42 V. PARALLEL LINES

Figure 38

opposite, then extending the side which is in the opposite direction we obtain an

angle supplementary to the ﬁrst angle, and equal to the second.

If both sides are in opposite directions, we extend both sides of the ﬁrst angle.

Thus we form an angle equal to the ﬁrst because they are vertical angles, and also

equal to the second.

Remark. Two corresponding angles and, therefore, two angles with sides par-

allel and in the same direction, have the same sense of rotation. We can therefore

say: Two angles with parallel sides are equal or supplementary, according as whether

they have the same sense of rotation or not.

Theorem. Two angles whose sides are perpendicular in pairs are equal or

supplementary, according as whether they have the same sense of rotation or not.

Figure 39

Consider angles BAC, B A C (Fig. 39) such that A B and A C are perpen-

dicular respectively to AB and AC. We draw AB1 perpendicular to AB, and turn

angle B1AC around on itself: side AB1 falls on AC, and hence line AB, perpen-

dicular to AB1, will occupy a position AC1 perpendicular to AC. We now have

an angle B1AC1 which is equal to BAC and has the same sense (because it was

constructed by reflection of angle CAB, whose sense is opposite to that of BAC)

and whose sides, perpendicular respectively to those of the ﬁrst angle, are therefore

parallel to those of B A C . Since the angles BAC, B A C are equal or supple-

mentary, according as their sides have the same or opposite senses of rotation, the

same is true of the given angles.