CHAPTER V
Parallel Lines
37. When two lines (Fig. 36) are intersected by a third (called a transversal),
this last line forms eight angles with the first two, which are numbered in the figure.
The relative positions of these angles are described as follows.
Figure 36
Two angles such as 3 and 5, situated between the two lines, and on different
sides of the transversal, are called alternate interior angles.
Two angles such as 3 and 6 situated between the two lines, but on the same
side of the transversal, are said to be interior on the same side.
Two angles such as 6 and 2 on the same side of the transversal, one between
the two lines, one outside, are said to be corresponding.
38. Definition. Two lines in the same plane are said to be parallel if they do
not intersect, no matter how far extended in either direction.
Theorem. Two lines intersected by the same transversal are parallel:
1◦.
If the interior angles on the same side are
supplementary;1
2◦.
If alternate interior angles are equal;
3◦.
If corresponding angles are equal.
1◦.
If the two lines were to intersect, on either side of the transversal, they
would form a triangle in which (25) the sum of two interior angles on the same side
would have to be less than two right angles.
1If
the angles 3 and 6 are supplementary, then so are 4 and 5, since the sum of these four
angles is four right angles.
39
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