Basic Mathematics for College Students

Vertical angles

• and


•  2 and  4

 1  3

9.1 Basic Geometric Figures; Angles 717

2 1 4

3

l 2

l 1

The Language of Mathematics When we work with two (or more) lines at

one time, we can use subscriptsto name the lines. The prefix submeans below

or beneath, as in submarine or subway. To name the first line in the figure

above, we use , which is read as “lsub one.” To name the second line, we use

l 2 , which is read as “lsub two.”

l 1

To illustrate that vertical angles always have the same measure, refer to the figure
below, with angles having measures of x,y, and 30°. Since the measure of any straight
angle is 180°, we have

To undo the addition of 30°,

subtract 30° from both sides.

Since xand yare both 150°, we conclude that xy.

x150° y150°

30 x180° and 30 y 180°

x

y

30 °

l 1

l 2

The previous example illustrates that vertical angles have the same measure.
Recall that when two angles have the same measure, we say that they are congruent.
Therefore, we have the following important fact.

Property of Vertical Angles

Vertical angles are congruent (have the same measure).

EXAMPLE (^3) Refer to the figure. Find:
a. b.
StrategyTo answer part a, we will use the property
of vertical angles. To answer part b, we will write an
equation involving that mathematically
models the situation.
WHYFor part a, we note that and intersect to
form vertical angles. For part b, we can solve the equation to find the unknown,
.
Solution
a.If we ignore for the moment, we see that and intersect to form the
pair of vertical angles and. By the property of vertical angles,
CBD   1 Read as “angle CBDis congruent to angle one.”

CBD  1

BC

·

AD

·

FE

·

m(ABF)

BC

·

AD

·

m(ABF)

m(1) m(ABF)

Self Check 3

Refer to the figure for Example 3.

Find:

a.

b.

Now TryProblems 69 and 71

m(DBE)

m(2)

Note that the angles

having measures xand y

are vertical angles.

100 °

50 °

(^12)
A
F B
E
D
C