Basic Mathematics for College Students

740 Chapter 9 An Introduction to Geometry

EXAMPLE (^3) In the figure, find.
StrategyWe will use the fact that the sum of the angle measures
of any triangle is 180° to write an equation that models the
situation.
WHYWe can then solve the equation to find the unknown angle measure,.
SolutionSince the sum of the angle measures of any triangle is 180°, we have
The symbol indicates that the measure of the
angle is 90°.
Do the addition.
To isolate x,undo the addition of 130° by
subtracting 130° from both sides.
Thus, is 50°.x
x50°
x130°180°



x40°90°180°

x

x

Self Check 3

In the figure, find .y

Now TryProblem 37

60 °

y

40 °

x

EXAMPLE (^4) In the figure, find the measure of each angle of.
StrategyWe will use the fact that the sum of the angle measures of any triangle
is 180° to write an equation that models the situation.
WHYWe can then solve the equation to find the
unknown angle measure , and use it to evaluate the
expressions and.
Solution
The sum of the angle measures
of any triangle is 180°.
Combine like terms: xx 2 x 4 x.
To isolate the variable term, 4x,
subtract 32° from both sides.
Do the subtractions.
To isolate x,divide both sides by 4.
Do the divisions. This is the measure of B.
To find the measures of and , we evaluate the expressions and
for.
Substitute 37 for x. Substitute 37 for x.
The measure of Bis 37°, the measure of Ais 69°, and the measure of Cis 74°.

69° 74°

x 32° 37 °32° 2 x2( 37 °)

2 x x37°

A C x32°

x37°

4 x

4



148°

4

4 x148°

4 x32°32°180°32°

4 x32°180°

x32°x 2 x180°

2 x x32°

x

ABC

Self Check 4

In ,the measure of

exceeds the measure of by

5°, and the measure of is

three times the measure of.

Find the measure of each angle

of.

Now TryProblem 41

DEF

E

F

E

DEF D

90

 40

130

B

A

C

x 2 x

x + 32°

37

4  148

 12

28

 28

0

18

7

0

10

 32

148

EXAMPLE (^5) If one base angle of an isosceles triangle measures 70°, what is
the measure of the vertex angle?
StrategyWe will use the isosceles triangle theorem and the fact that the sum of
the angle measures of any triangle is 180° to write an equation that models the
situation.
WHYWe can then solve the equation to find the unknown angle measure.
Self Check 5
If one base angle of an isosceles
triangle measures 33°, what is the
measure of the vertex angle?
Now TryProblem 45