Basic Mathematics for College Students

Some geometric figures contain two transversals.

9.2 Parallel and Perpendicular Lines 729

EXAMPLE (^2) Refer to the figure. If
and , find the measures of the
other seven angles that are labeled.
StrategyWe will look for vertical angles,
supplementary angles, and alternate interior
angles in the figure.
WHYThe facts that we have studied about
vertical angles, supplementary angles, and alternate interior angles enable us to use
known angle measures to find unknown angle measures.
Solution
3 and 1 are supplementary: m(3) m(1) 180°.
Vertical angles are congruent: m(2) m(3).
Vertical angles are congruent: m(4) m(1).
If two parallel lines are cut by a transversal, alternate interior angles
are congruent: m(5) m(4).
If two parallel lines are cut by a transversal, alternate interior angles
are congruent: m(6) m(3).
Vertical angles are congruent: m(7) m(6).
m(8)60° Vertical angles are congruent: m(8) m(5).
m(7)120°
m(6)120°
m(5)60°
m(4)60°
m(2)120°
m(1)60°
m(3)120°
l 1 l 2
Self Check 2
Refer to the figure for Example 2.
If and , find
the measures of the other seven
angles that are labeled.
Now TryProblem 23
l 1 l 2 m(8)50°
56
78
3 4
1 2
l 3
l 2
l 1
EXAMPLE (^3) Refer to the figure. If ,
which pairs of angles are congruent?

ABDE

Self Check 3

See the figure below. If ,

which pairs of angles are

congruent?

YZMN

Now TryProblem 25

3 4

1 2

C

DE

A B

M

X

N

Z

Y

(^21)
3
4
EXAMPLE (^4) In the figure,. Find.
StrategyWe will use the corresponding angles
property to write an equation that mathematically
models the situation.
WHYWe can then solve the equation to find.
SolutionIn the figure, two corresponding angles have degree measures that are
represented by the algebraic expressions and. Since , this
pair of corresponding angles are congruent.
9 x15° 6 x30° l 1 l 2
x
l 1 l 2 x
Self Check 4
In the figure below,l 1 l 2. Find .y
Now TryProblem 27
StrategyWe will use the corresponding angles
property twice to find two pairs of congruent angles.
WHYBoth and are transversals cutting the
parallel line segments ABand DE.

BC

·

AC

·

SolutionSince , and is a transversal cutting them, corresponding
angles are congruent. So we have

Since and is a transversal cutting them, corresponding angles
must be congruent. So we have

B   2

BC

·

ABDE

A   1

AC

·

ABDE

9 x − 15 °

6 x + 30 °

l 2

l 1

7 y − 14 °

4 y + 10 °

l 2

l 1