3.3: Angles of Polygons
S
What is a Polygon?
S
A polygon is a closed plane figure made up of three or more line segments that intersect
only at their endpoints.
S
In a regular polygon all the sides are congruent, and all of the interior angles are
congruent.
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Interior angle measures of a polygon:
S The sum S of the interior angle measures of a polygon with n sides is
S  (n  2)  180

0
Concave and Convex Polygons
S
A polygon is convex when every
line segment connecting any two
vertices lies entirely inside the
polygon.
S
A polygon is concave when at
least one line segment connecting
any two vertices lies outside the
polygon.
Finding the Sum of Interior
Angle Measures
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Find the sum of the interior angle
measures of the school crossing
sign.
S
The sign is in the shape of a
pentagon. It has 5 sides.
S  (n  2)  180
0
S  (5  2)  180
0
S  3  180
S  540
0
0
Practice
Find the sum of the interior angle measures of the green polygon.
S
The web has 7 sides
S
The honey comb has 6 sides
S  ( n  2)  180
0
S  (n  2)  180
0
S  ( 7  2)  180
0
S  (6  2)  180
0
S  5  180
S  900
0
0
S  4  180
S  720
0
0
Finding an Interior Angle
Measure of a Polygon
Find the value of x.
S
Step 1: The polygon has 7 sides.
Find the sum of the interior
angle measures.
S  ( 7  2)  180
S  5  180
0
0
0
S  900
S Step 2: Write and solve an
equation to find x.
140  145  115  120  130  128  x  900

778  x  900
x  122
0
Practice
Find the value of x.
S  (6  2)  180
S  4  180
S  720
0
0
S  ( 4  2)  180
S  2  180
0
S  360
0
0
0
90  80  115  x  360
615  x  720
285  x  360

x  105
0

x  75
S  3  180
S  540
135  110  125  120  125  x  720
0
S  (5  2)  180

0
0
0
145  145  110  2 x  2 x  540
400  4 x  540
4 x  140
x  35
0
Real Life Application
S
S
A cloud system discovered on
Saturn is in the approximate
shape of a regular hexagon. Find
the measure of each interior
angle of the hexagon.
Step 1: A hexagon has 6 sides.
Find the sum of the interior
angle measures.
S  (n  2)  180
0
S  (6  2)  180
0
S  4  180
S  720
S

0
0
Step 2: Divide the sum by the
number of interior angles, 6.
720
0
 6  120
0
Exterior Angle Measures of a
Polygon
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The sum of the measures of the
exterior angles of a convex
polygon is 3600.
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Algebra
S w+x+y+z=3600
Finding Exterior Angle
Measures
Find the measures of the exterior angles of each polygon.
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Write and solve an equation for x
S
Write and solve an equation for z
x  50  127  91  360
124  z  z  26  360
x  268  360
2 z  150  360
x  92
0
2 z  210
z  105
0
The exterior
angles
measures are
1050 and
(105+26)=13
10
Practice
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Find the measures of the interior
angles of the polygon.
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Write and solve an equation to
find x.
90  90  90  x  x  360
270  2 x  360
2 x  90
x  45
0
The exterior angle measures are 900, 450, 900, 900, and 450.