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3.5 The Polygon
Angle-Sum Theorems
Chapter 3: Parallel and
Perpendicular Lines
Goals / “I can…”
Classify polygons
Find the sum of the measures of interior and
exterior angles of polygons
Polygon:
A closed plane figure.
w/ at least 3 sides (segments)
The sides only intersect at their endpoints
Name it by starting at a vertex & go around
the figure clockwise or counterclockwise
listing each vertex you come across.
3.5 The Polygon Angle-Sum
Theorems
Polygon: a closed plane figure with at least
three sides that are segments
A polygon
Not a polygon;
Not enclosed
Not a polygon;
Two sides intersect
Naming a Polygon
Name a polygon by its vertices.
A
ABCDE or AEDCB
B
E
C
D
Start at one vertex and go around in order
Naming a Polygon
Three polygons are pictured. Name each polygon:
L
P
M
O
N
Classifying a Polygon by the
number of sides:
Sides
3
4
5
6
7
8
9
10
12
n
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
Convex vs. Concave
A Convex Polygon has all vertices pointing “out”
A Concave Polygon has one or more vertices
“caving in”
Classify
Classify each polygon by its sides. Identify each as
convex or concave:
Hexagon; Convex
Octagon; Concave
III.
Polygon Interior ∠ sum
4 sides
2 ∆s
2 • 180 = 360
5 sides
3 ∆s
3 • 180 = 540
Sum of Polygon Angle
Measures
Use triangles to figure out the sum of the angles in each
polygon:
# of Sides:
# of Triangles:
Total Degrees:
# of Sides:
# of Triangles:
Total Degrees:
6 sides
4 ∆s
4 • 180 = 720
• All interior ∠ sums
are multiple of 180°
Th(3-9) Polygon Angle – Sum Thm
Sum of Interior ∠
S = (n -2) 180
# of sides
Sum of Polygon Angle
Measures
Number of Sides
3
4
5
6
n
Number of
Triangles
1
Total Degrees
inside Polygon
180
Theorem 3-9
Polygon Angle Sum Theorem
The sum of the measures of the angles in a polygon
is (n – 2)180.
Find the sum of the measure of the angles of a 15-gon.
Polygon Angle Sum
The sum of the measures of the angles of a given polygon
is 720. How many sides does the polygon have?
Use (n – 2)180 :
Examples 2 & 3:
Find the interior ∠
sum of a 15 – gon.
S = (n – 2)180
S = (15 – 2)180
S = (13)180
S = 2340
Find the number of
sides of a polygon if it
has an ∠ sum of 900°.
S = (n – 2)180
900 = (n – 2)180
5=n–2
n = 7 sides
Using Polygon Angle-Sum
Theorem
Find the measure of <Y in pentagon TVYMR at the right.
R
135°
M
T
Use (n – 2)180
90°
Y
V
Write an equation to solve for <Y
Using Polygon Angle-Sum
Theorem
Pentagon ABCDE has 5 congruent angles. Find the
measure of each angle.
Use the Polygon Angle-Sum Theorem: (n – 2)180
Divide the total number of degrees by the number of angles:
Exterior Angles
What do you notice about each set of exterior angles?
80°
75°
115°
2
1
150°
99°
130°
71°
70°
86°
88°
1:
3
2:
70°
46°
3:
Theorem 3-10 Polygon AngleSum Theorem
The sum of one set of exterior angles for any polygon is
360°.
1
5
2
4
3
m<1 + m<2 + m<3 + m<4 + m<5 = 360°
Polygons
Equilateral Polygon: all sides congruent
Equiangular Polygon: all angles congruent
Regular Polygon: all sides and all angles congruent
(equiangular and equilateral)
*If a polygon is a regular polygon then all of the exterior
angles are also congruent.
Example 4:
How many sides does a polygon have if it has
an exterior ∠ measure of 36 .
= 36
360 = 36n
10 = n
Example 5:
Find the sum of the interior ∠s of a polygon if it
has one exterior ∠ measure of 24.
360 = 24
n
n = 15
S = (n - 2)180
= (15 – 2)180
= (13)180
= 2340
Example 6:
Solve for x in the following example.
x
4 sides
Total sum of interior ∠s = 360
100
90 + 90 + 100 + x = 360
280 + x = 360
x = 80
Example 7:
Find the measure of one interior ∠ of a regular
hexagon.
S = (n – 2)180
= (6 – 2)180
= (4)180
= 720
720
=
6
= 120