Figure 37
This is true because the perpendicular must intersect the other line (40, Corol-
lary II) and the angle it forms will be a right angle, because of the theorem we have
just proved.
Remarks. I. The corresponding equal angles xEB and xF D have the same
direction of rotation.
The two parallel directions EB, F D, both situated on the same side of a
common transversal, are called parallel and in the same direction.
II. We have used a new method of proof of a converse, different from the ones
used in 32, and which consists in establishing the converse with the help of the
original theorem. It should be noted here that the argument basically depends on
the fact that the parallel through E to CD is unique.
42. According to the theorem of 38 and its converse, the definition of parallels
amounts to the following: Two lines are parallel if they form, with an arbitrary
transversal, equal corresponding angles (or equal alternate interior angles, or sup-
plementary interior angles on the same side of the transversal).
This definition, equivalent to the first, is usually more convenient to use.
In place of the phrase parallel lines we often substitutes lines with the same
direction, whose meaning is clear from the preceding propositions.
Remark. Because of what we have just noted, two lines which coincide must
be viewed as a particular case of two parallel lines.
43. Theorem. Two angles with two pairs of parallel sides are either equal or
supplementary. They are equal if the sides both lie in the same direction or both
in opposite directions; they are supplementary if a pair of sides lie in the same
direction, and the other pair opposite.
First, two angles with a common side, and whose second sides are parallel in
the same direction (Fig. 38) are equal because they are corresponding angles. Two
angles whose sides are parallel and in the same direction are then equal because
one side of the first angle and one side of the second angle will form a third angle
equal to the first two. If one of the sides is in the same direction, with the other
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