Lesson 6-1 Angles of Polygons
TARGETS
• 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.
LESSON 6.1: Angles of Polygons
Polygon Interior Angles Sum
# of Sides Angle Measure
mA=
mC=
3
mB=
mF=
mH=
4
mG=
mI=
mP=
mS=
5
mQ=
mT=
mR=
Sum of Angles
LESSON 6.1: Angles of Polygons
Polygon Interior Angles Sum
The sum of the interior angle
measures of an n-sided convex
polygon is 180(n - 2)
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LESSON 6.1: Angles of Polygons
EXAMPLE 1
Find the Interior Angles Sum of a Polygon
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
= 7 ● 180 or 1260
n=9
Simplify.
Answer: The sum of the measures is 1260.
LESSON 6.1: Angles of Polygons
EXAMPLE 1
Find the Interior Angles Sum of a Polygon
B. Find the measure of each interior angle of
parallelogram RSTU.
Step 1 Find x.
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.
Find the Interior Angles Sum of a Polygon
Sum of measures
of interior angles
Substitution
Combine like
terms.
Subtract 8 from
each side.
Divide each side
by 32.
Find the Interior Angles Sum of a Polygon
Step 2 Use the value of x to find the measure of
each angle.
m  R = 5x
= 5(11) or 55
m  S = 11x + 4
= 11(11) + 4 or 125
m  T = 5x
= 5(11) or 55
m  U = 11x + 4
= 11(11) + 4 or 125
Answer: mR = 55, mS = 125, mT = 55,
mU = 125
LESSON 6.1: Angles of Polygons
EXAMPLE 2 Interior Angle Measure of Regular Polygon
Find the measure of each interior angle of a regular
octagon.
LESSON 6.1: Angles of Polygons
EXAMPLE 3
Find Number of Sides Given Interior Angle Measure
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)
(150)n = 180(n – 2)
150n = 180n – 360
0 = 30n – 360
Interior Angle Sum
Theorem
S = 150n
Distributive Property
Subtract 150n from each
side.
Find Number of Sides Given Interior Angle
Measure
360 = 30n
12 = n
Add 360 to each side.
Divide each side by 30.
Answer: The polygon has 12 sides.
LESSON 6.1: Angles of Polygons
Polygon Exterior Angles Sum
The sum of the exterior angle
measures of a convex polygon, one
angle at each vertex, is 360
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LESSON 6.1: Angles of Polygons
EXAMPLE 4 Find Exterior Angle Measures of a Polygon
A. Find the value of x in the diagram.
Find Exterior Angle Measures of a Polygon
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
LESSON 6.1: Angles of Polygons
EXAMPLE 4
Find Exterior Angle Measures of a Polygon
B. Find the measure of each exterior angle of a
regular decagon.
A regular decagon has 10 congruent sides and 10
congruent angles. The exterior angles are also
congruent, since angles supplementary to congruent
angles are congruent. Let n = the measure of each
exterior angle and write and solve an equation.
10n = 360
Polygon Exterior Angle
Sum Theorem
n = 36
Divide each side by 10.
Answer:
The measure of each exterior angle of a
regular decagon is 36.