728 Chapter 9 An Introduction to Geometry
Solution
a.To identify corresponding angles, we examine the angles to the right of the
transversal and the angles to the left of the transversal. The pairs of
corresponding angles in the figure are- and • and
- and • and
b.To identify the interior angles, we determine the angles inside the two lines cut
by the transversal. The interior angles in the figure are
, , , and
c.Alternate interior angles are nonadjacent angles on opposite sides of the
transversal inside the two lines. Thus, the pairs of alternate interior angles in
the figure are - 3 and 5 • 4 and 6
3 4 5 6
2 6 3 7
1 5 4 8
3 Use properties of parallel lines cut by a transversal
to find unknown angle measures.
Lines that are cut by a transversal may or may not be parallel. When a pair of parallel
lines are cut by a transversal, we can make several important observations about the
angles that are formed.- Corresponding angles property:If two parallel lines are cut by a transversal,
each pair of corresponding angles are congruent. In the figure below, if ,
then , , , and. - Alternate interior angles property:If two parallel lines are cut by a transversal,
alternate interior angles are congruent. In the figure below, if , then
and. - Interior angles property:If two parallel lines are cut by a transversal, interior
angles on the same side of the transversal are supplementary. In the figure
below, if l 1 l 2 , then 3 is supplementary to 5 and 4 is supplementary to 6.
3 6 4 5
l 1 l 21 5 3 7 2 6 4 8
l 1 l 234
125678l 3l 2l 1Transversall 1 l 2l 3l 2l 1(a)- If a transversal is perpendicular to one of two parallel lines, it is also
perpendicular to the other line. In figure (a) below, if and , then. - If two lines are parallel to a third line, they are parallel to each other. In figure
(b) below, if l 1 l 2 and l 1 l 3 , then l 2 l 3.
l 1 l 2 l 3 ⊥ l 1 l 3 ⊥ l 2l 2l 3l 1(b)