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8.1 Angles of Polygons
Objectives

Find the sum of the measures of the
interior angles of a polygon

Find the sum of the measures of the
exterior angles of a polygon
Sum of the Measures of the Interior
Angles of a Polygon
We have already learned the name of a
polygon depends on the number of sides in
the polygon: triangle, quadrilateral,
pentagon, hexagon, and so forth.
The sum of the measures of the interior angles
of a polygon also depends on the number of
sides.
Sum of the Measures of the Interior
Angles of a Polygon
From a previous lesson we learned the sum of
the measures of the interior angles of a
quadrilateral are known by dividing the
quadrilateral into two triangles.
You can use this triangle method to find the
sum of the measures of the interior angles of
any convex polygon with n sides, called an
n - gon.
Sum of the Measures of the Interior
Angles of a Polygon
Polygon
Triangle
# of
sides
3
Sum of measures of
interior ’s
1
1 ● 180 = 180
2 ● 180 = 360
Quadrilateral
Pentagon
Hexagon
Nonagon (9)
n - gon
# of
triangles
n
Sum of the Measures of the Interior
Angles of a Polygon

From the previous slide, we have discovered
that the sum of the measures of the interior
angles of a convex n - gon is
(n – 2) ● 180

This relationship can be used to find the
measure of each interior angle in a regular
n - gon because the angles are all congruent.
Interior Angle Sum Theorem

Interior Angle Sum Theorem
If a convex polygon has n sides and
S is the sum of its interior angles,
then S = 180(n – 2).
Example 1:
ARCHITECTURE
A mall is designed so that
five walkways meet at a
food court that is in the
shape of a regular
pentagon. Find the sum of
measures of the interior
angles of the pentagon.
Since a pentagon is a convex polygon, we can use the
Angle Sum Theorem.
Example 1:
Interior Angle Sum Theorem
Simplify.
Answer: The sum of the measures of the angles is 540.
Your Turn:
A decorative window is designed to have the shape of
a regular octagon. Find the sum of the measures of the
interior angles of the octagon.
Answer: 1080
Example 2:
The measure of an interior angle of a regular polygon
is 135. 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.
Interior Angle Sum Theorem
Distributive Property
Subtract 135n from each side.
Add 360 to each side.
Divide each side by 45.
Answer: The polygon has 8 sides.
Your Turn:
The measure of an interior angle of a regular polygon
is 144. Find the number of sides in the polygon.
Answer: The polygon has 10 sides.
Example 3:
Find the measure of each interior angle.
Since
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.
Example 3:
Sum of measures of
angles
Substitution
Combine like terms.
Subtract 8 from each
side.
Divide each side by
32.
Example 3:
Use the value of x to find the measure of each angle.
Answer:
Your Turn:
Find the measure of each interior angle.
Answer:
Sum of the Measures of the Exterior
Angles of a Polygon
Interestingly, the measures of the exterior angles of a
polygon is an even easier formula. Let’s look at the
following example to understand it.
Exterior Angle Sum Theorem
Exterior Angle Sum Theorem
If a polygon is convex, then
the sum of the measures of
the exterior angles, one at
each vertex, is 360°.
Example 4:
Find the measures of an exterior angle and an interior
angle of convex regular nonagon ABCDEFGHJ.
At each vertex, extend a side to form one exterior angle.
Example 4:
The sum of the measures of the exterior angles is 360. A
convex regular nonagon has 9 congruent exterior angles.
Divide each side by 9.
Answer: The measure of each exterior angle is 40. Since
each exterior angle and its corresponding
interior angle form a linear pair, the measure of
the interior angle is 180 – 40 or 140.
Your Turn:
Find the measures of an exterior angle and an interior
angle of convex regular hexagon ABCDEF.
Answer: 60; 120
Assignment

Pre-AP Geometry:
Pg. 407 #14 - 40

Geometry:
Pg. 407 #4 – 15, 21 – 24,
27 – 28, 32, 35, 36