6-1 Angles of Polygons
You named and classified polygons.
• Find and use the sum of the measures of the
interior angles of a polygon.
• Find and use the sum of the measures of the
exterior angles of a polygon.
Definitions

A polygon is a plane figure whose sides
are three or more coplanar segments that
intersect only at their endpoints (the
vertices). Consecutive sides cannot be
collinear, and no more than two sides can
meet at any one vertex.
Definition

A diagonal of a polygon is a line
segment whose endpoints are any two
nonconsecutive vertices of the polygon.
diagonal
Classification of Polygons
3 sides Triangle
4 sides Quadrilateral
5 sides Pentagon
6 sides Hexagon
7 sides Heptagon
8 sides Octagon
9 sides Nonagon
10 sides Decagon
12 sides Dodecagon
20 sides Icosagon
Polygon interior angles
What happens to the sum of the degrees of
the inside angles as the number of sides
increases?
180°
360°
540°
720°
Is there an easier way to figure out the
sum of the degrees of the inside angles
for any polygon?
Angle-Sum Theorem for Polygons
The sum of the measures of the interior
angles of a convex polygon with n sides is
given by S = (n − 2)180°
Page 393
A. Find the sum of the measures of the
interior angles of a convex nonagon.
A nonagon has nine sides. Use the Polygon
Interior Angles Sum Theorem to find the sum
of its interior angle measures.
(n – 2) ● 180= (9 – 2) ● 180
n=9
= 7 ● 180 or 1260 Simplify.
Answer: The sum of the measures is 1260.
B. Find the measure of
each interior angle of
parallelogram RSTU.
Step 1 Find x.
Since n=4, the sum of the measures of the interior
angles is
Write an equation to
express the sum of the measures of the interior
angles of the polygon.
Sum of measures of
interior angles
Substitution
Combine like terms.
Subtract 8 from each side.
Divide each side by 32.
Step 2 Use the value of x to find the measure
of
each angle.
mR = 5x
mS = 11x + 4
= 5(11) or 55
= 11(11) + 4 or 125
mT = 5x
mU = 11x + 4
= 5(11) or 55
= 11(11) + 4 or 125
Answer: mR = 55, mS = 125, mT = 55,
mU = 125
A. Find the sum of the measures of the
interior angles of a convex octagon.
A. 900
B. 1080
C. 1260
D. 1440
A pottery mold makes bowls that are in the
shape of a regular heptagon. Find the
measure of one of the interior angles of the
bowl.
A. 130°
B. 128.57°
C. 140°
D. 125.5°
The sum of the measures of the
interior angles of a convex polygon
is 900°. Find the number of sides of
the polygon.
S = (n− 2)180°
900 = (n − 2)180°
900÷180 = (n − 2)180° ÷180
5=n−2
5+2=n−2+2
7=n
The measure of an interior angle of a regular
polygon is 150. Find the number of sides in
the polygon.
Use the Interior Angle Sum Theorem to write an
equation to solve for n, the number of sides.
S = 180(n – 2)
Interior Angle Sum
Theorem
(150)n = 180(n – 2)
150n = 180n – 360
0 = 30n – 360
360 = 30n
12 = n
S = 150n
Distributive Property
Subtract 150n from each
side.
Add 360 to each side.
Divide each side by 30.
Answer: The polygon has 12 sides.
The measure of an interior angle of a regular
polygon is 144. Find the number of sides in
the polygon.
A. 12
B. 9
C. 11
D. 10
Polygon exterior angles
What happens to the sum of the degrees of
the outside angles as the number of sides
increases?
360°
360°
360°
360°
Is there an easier way to figure out the
sum of the degrees of the outside angles
for any polygon?
Exterior Angle Theorem for
Polygons
The sum of the measures of the exterior
angles of a convex polygon (one at each
vertex) is 360°
Page 396
Find the sum of the interior angles
AND exterior angles for the polygon
7 sides Heptagon
Interior angle sum = (n − 2)180°
= (7 − 2)180°
= (5)180°
= 900°
Exterior angle sum = 360°
Find the sum of the interior angles
AND exterior angles for the polygon
22-gon
Interior angle sum = (n − 2)180°
= (22 − 2)180°
= (20)180°
= 3600°
Exterior angle sum = 360°
A. Find the value of x
in the diagram.
Use the Polygon
Exterior Angles Sum
Theorem to write an
equation. Then solve
for x.
5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x +
3) +(5x + 5) = 360
(5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 +
(–12) + 3 + 5] = 360
31x – 12 = 360
31x = 372
x = 12
Answer: x = 12
B. Find the measure of each exterior angle
of a regular pentagon.
A. 72
B. 60
C. 45
D. 90

How do you find the sum of the measures of
the interior angles of a convex polygon?
S = (n −2)180°
How do you find the measure of one of the
interior angles of a convex polygon?
S = (n −2)180°/n

What is the sum of the measure of the
exterior angles of a convex polygon?
 360°
 How do you find the measure of one of the
exterior angles of a convex polygon?
 360°/n
6-1 Assignment
Page 398, 13-37 odd