42 V. PARALLEL LINES
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.
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.