Basic Mathematics for College Students

9.6 Quadrilaterals and Other Polygons 771

Self Check 4

Refer to the isosceles trapezoid

shown below with. Find

xand .y

RSUT

Now TryProblem 31

x 10 in.

58° y

U T

RS

4 Use the formula for the sum of the

angle measures of a polygon.

In the figure shown below, a protractor was used to find the measure of each angle of
the quadrilateral. When we add the four angle measures, the result is 360°.

88°

88° + 79° + 127° + 66° = 360°

79°

127°

66°

This illustrates an important fact about quadrilaterals: The sum of the measures
of the angles of anyquadrilateral is 360°. This can be shown using the diagram in figure
(a) on the following page. In the figure, the quadrilateral is divided into two triangles.
Since the sum of the angle measures of any triangle is 180°, the sum of the measures
of the angles of the quadrilateral is , or 360°.
A similar approach can be used to find the sum of the measures of the angles of
any pentagon or any hexagon. The pentagon in figure (b) is divided into three
triangles. The sum of the measures of the angles of the pentagon is , or 540°.
The hexagon in figure (c) is divided into four triangles. The sum of the measures of the
angles of the hexagon is , or 720°. In general, a polygon with sides can be
divided into triangles. Therefore, the sum of the angle measures of a polygon can
be found by multiplying 180° by n 2.

n 2

4 180° n

3 180°

2 180°

EXAMPLE (^4) Landscaping A cross section of a drainage ditch shown
below is an isosceles trapezoid with ABDC. Find and .x y
StrategyWe will compare the nonparallel sides and compare a pair of base angles
of the trapezoid to find each unknown.
WHYThe nonparallel sides of an isosceles trapezoid have the same length and
both pairs of base angles are congruent.
SolutionSince and are the nonparallel sides of an isosceles trapezoid,
and are equal, and is 8 ft.
Since and are a pair of base angles of an isosceles trapezoid, they are
congruent and m(D)m(C). Thus, is 120°.y

D C

m(AD) m(BC) x

AD BC

A

D C

120°

8 ft x

B

y

2

8

3

8

79

127

 66

360