Basic Mathematics for College Students

9.3 Triangles 741

SolutionBy the isosceles triangle theorem, if one of the base angles
measures 70°, so does the other. (See the figure on the right.) If we
let represent the measure of the vertex angle, we have

The sum of the measures of the angles of a

triangle is 180°.

Combine like terms: 70°70°140°.

To isolate x, undo the addition of 140° by

subtracting 140° from both sides.

The vertex angle measures 40°.

x40°

x140°180°

x70°70°180°

x

70 ° 70 °

x

EXAMPLE (^6) If the vertex angle of an isosceles triangle measures 99°, what
are the measures of the base angles?
StrategyWe will use the fact that the base angles of an isosceles triangle have the
same measure and the sum of the angle measures of any triangle is 180° to write an
equation that mathematically models the situation.
WHYWe can then solve the equation to find the unknown angle measures.
SolutionThe base angles of an isosceles triangle
have the same measure. If we let represent the
measure of one base angle, the measure of the other
base angle is also. (See the figure to the right.) Since
the sum of the measures of the angles of any triangle is 180°, the sum of the
measures of the base angles and of the vertex angle is 180°. We can use this fact to
form an equation.
Combine like terms: xx 2 x.
To isolate the variable term, 2x,undo the addition
of 99° by subtracting 99° from both sides.
To isolate x,undo the multiplication by 2 by
dividing both sides by 2.
The measure of each base angle is 40.5°.
x40.5°
2 x
2



81°

2

2 x81°

2 x99°180°

xx99°180°

x

x

Self Check 6

If the vertex angle of an isosceles

triangle measures 57°, what are the

measures of the base angles?

Now TryProblem 49

99 °

xx

1. a. 4 b. 5 2.no 3.30° 4. ,,
   5.114° 6.61.5°

m(D)40°m(E)35°m(F)105°

ANSWERS TO SELF CHECKS

SECTION 9.3 STUDY SET

VOCABULARY

Fill in the blanks.

1. A is a closed geometric
   figure with at least three line
   segments for its sides.


2. The polygon shown to the right has
   seven and seven vertices.
   3. A point where two sides of a polygon intersect is
   called a of the polygon.
   4. A polygon has sides that are all the same
   length and angles that all have the same measure.
   5. A triangle with three sides of equal length is called an
   triangle. An triangle has at least
   two sides of equal length. A triangle has no
   sides of equal length.

40.5

(^2)  81.0
 8
01
 0
10
 10
0