Download 3.4 The Polygon Angle

The Polygon AngleSum Theorems
3.4 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:
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:
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 :
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.