Basic Mathematics for College Students

718 Chapter 9 An Introduction to Geometry

Since congruent angles have the same measure,

In the figure, we are given. Thus, is also 50°, and we

can write.

b.Since is a straight angle, the sumof the measures of , the 100°

angle, and the 50° angle is 180°. If we let , we have

The word sumindicates addition.

On the left side, combine like terms: 100°50°150°.

To isolate x,undo the addition of 150° by subtracting

150° from both sides: 180°150°30°.

Thus,m(ABF)30°

x30°

x150°180°

x100°50°180°

xm(ABF)

ABD ABF

m(1)50°

m(CBD)50° m(1)

m(CBD)m(1)

EXAMPLE (^4) In the figure on the right, find:
a.x b. c.
StrategyWe will use the property of vertical
angles to write an equation that mathematically
models the situation.
WHY and intersect to form two pairs of vertical angles.
Solution
a.In the figure, two vertical angles have degree measures that are represented by
the algebraic expressions and. Since the angles are vertical
angles, they have equal measures.
Set the algebraic expressions equal.
To eliminate 3xfrom the right side, subtract
3 xfrom both sides.
Combine like terms: 4x 3 xx
and 3x 3 x0.
To isolate x,undo the subtraction of
20° by adding 20° to both sides.
Thus, is 35°.
b.To find , we evaluate the expression for.
Substitute 35° for x.
Do the multiplication.
Do the addition.
Thus,.
c. is a straight angle. Since the measure of a straight angle is 180° and
m(ABC)120°,m(CBE)must be 180°120°, or 60°.

ABE

m(ABC)120°

120°

105°15°

3 x15°3( 35 )15°

m(ABC) 3 x15° x35°

x

x35°

x20°15°

4 x20° 3 x 3 x15° 3 x

4 x20° 3 x15°

4 x20° 3 x15°

DC

·

AE

·

m(ABC) m(CBE)

Self Check 4

In the figure below, find:

a.y

b.

c.m(MYX)

m(XYZ)

A

D

C

E

B

3 x + 15 °

4 x − 20 °

4 y − 10 ° 2 y + 20 °

X

Z

M

N

Y

Now TryProblem 75

7 Solve problems involving complementary

and supplementary angles.

Complementary and Supplementary Angles

Two angles are complementary angleswhen the sum of their measures is 90°.

Two angles are supplementary angleswhen the sum of their measures is 180°.

3

1

5

 3

105