3.5: The Polygon Angle-Sum Theorems
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Classifying Polygons
Riddle: What did the owner say when he lost his parrot?
Classifying Polygons
Polygon: Closed plane figure with at least three sides that are
segments intersecting only at their endpoints, and no adjacent
sides are collinear.
B
B
C
A
A
B
C
C
A
E
D
Not a polygon
D
E
Not a polygon
D
E
Polygon
Classifying Polygons
Note: A diagonal of a polygon is a segments that connects
two nonconsecutive vertices.
Convex Polygon: A polygon with no diagonal with points
outside the polygon
Concave Polygon: A polygon with at least one diagonal with
points outside the polygon.
Convex
Concave
Classifying Polygons
We can classify polygons according to the number of sides it has.
Sides
Name
3
Triangle
Quadrilateral
Pentagon
Hexagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
4
5
6
8
9
10
12
n
Classifying Polygons
Classify each polygon by its sides. Identify each as
convex or concave.
Classifying Polygons
Equilateral Polygon: A polygon
with all sides congruent.
Equiangular Polygon: A polygon
with all angles congruent.
Regular Polygon: A polygon that is both equilateral and
equiangular.
Naming Polygons
G
D
H
K
Polygon:
Vertices:
Sides:
Angles:
J
Super Fun Activity!
 Sketch polygons with 4, 5, 6, 7, and 8 sides
 Divide into triangles by drawing diagonals from one vertex
 Multiply the # of Δ’s by 180 to find the polygon angle-sum
# of Sides
# of Δ’s Formed
Angle-Sum
4
5
6
7
8
Theorem 3-14: Polygon Angle-Sum Theorem: The sum of the
measures of the angles in an n-gon is (n – 2)180.
Polygon Angle Sums
Find the sum of the measures of the angles of a regular
dodecagon. Then find the measure of an interior angle.
The sum of the angles of a polygon is 720. Find the number
of sides the polygon has and classify it.
Polygon Angle Sums
Find x in the following polygons.
125°
x°
125°
95°
102°
94°
171°
x°
97°
Polygon Angle Sums
Theorem 3-15: Polygon Exterior Angle-Sum Theorem: The
sum of the measures of the exterior angles of a polygon, one at
each vertex, is 360.
For a pentagon: m1 m2  m3  m4  m5  360
3

2
4
1
5
Polygon Angle Sums
If the measure of an exterior angle of a regular polygon is 30, find
the number of sides and classify it.
3.5: The Polygon Angle-Sum Theorems
HOMEWORK: 3.5: #2-26 even, 32, 34, 40,42
THT: 1-11
TERMS: Polygon, Convex Polygon, Concave Polygon,
Equilateral Polygon, Equiangular Polygon, Regular Polygon
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