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