Download Chapter 3 Parallel Lines and Planes

3-4: The polygon
Angle-Sum
Theorems
Polygon means many angles.
• A polygon is a plane figure that
meets the following conditions.
1)It is formed by 3 or more segments
called sides, such that no two sides
with a common endpoint are
collinear.
2) Each side intersects exactly two
other sides, one at each vertex.
Each endpoint of a side is a
VERTEX of the polygon.
VERTICES is plural of vertex.


You can name a
polygon by listing
its vertices
consecutively.
Polygon PQRST or
QPTSR etc.
R is a vertex.
RS is a side.
Q
R
P
T
S
Polygon
Polygon
Not a
Polygon
Not a
Polygon
Not a
Polygon
p 322
# of sides
3
4
5
6
7
8
9
10
12
n
name
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
convex
A __________
polygon is a polygon
where when extended, no sides
intersect the interior of the polygon.
convex
concave
Ex 1. Is the figure a convex polygon,
concave polygon, or neither?

A)

B)
Concave polygon
Convex polygon

C)

D)
Not a polygon
Not a polygon
Different Examples of Shapes!
Polygon
# of Sides
# of
s
Sum of Measures
of interior
s
Triangle
3
1
1(180)=180 °
Quad
4
2
2(180)=360 °
Pentagon
5
3
3(180)=540 °
Hexagon
6
4
4(180)=720 °
N-gon
n
(n-2)
(n-2)180°
Theorem 3-9: Polygon
Angle-Sum Theorem
The sum of the measures of
the interior angles of a convex
n-gon is:
Sum = (n-2)180°
**Where n represents the number of sides
of a polygon**
Example 1- Find the sum of the measure
of the angles of an:
I  (n  2)180
a.) Octagon
I  (8  2)180
I  (6)180
I  1080
b.)15-gon
I  (n  2)180
I  (15  2)180
I  (13)180
I  2340
Example 2
Solve for x
I  (n  2)180
I  (6  2)180
88°
136°
105°
136°
I  (4)180
I  720
142°
x°
The sum of all the other
angles.
x  720  607  113
Theorem 3-10
The sum of the measures of the
exterior angles of a convex polygon,
3
one at each vertex, is
2
4
1
Sum=360°
5
m1  m2  m3  m4  m5  360
Example 3- Find the sum of the exterior
angles of aa.) Pentagon
E  360
b.) Dodecagon
E  360
•A polygon is EQUILATERAL if all
of its sides are congruent.
* A polygon is EQUIANGULAR if
all of its interior angles are
congruent.
•A polygon is REGULAR if it is
equilateral and equiangular.
Regular Pentagon

The measure of each angle of a regular n-gon
is
Each Interior Angle =(n-2)180
n
**where n represents the number of sides
of the polygon**
Example 4- Find the measures of
each angle of a regular
hexagon.
(n  2)180
i
n
(6  2)180
i
6
4(180)
i
6
720
i
6
i  60
(More) Theorems!
The measure of each exterior angle of a
regular n-gon is
e=360°
n=360°
n
e
**n represents the number of sides of the polygon and e represents the
measure of each exterior angle.
i  e  180
**i represents the measure of an interior angle and
e represents the measure of an exterior angle.
Example 5- The measure of each interior
angle of a regular polygon is 140°.
How many sides does the polygon
have?
i  e  180
140  e  180
e  40
360
n
e
360
n
40
n=9 sides (Nonagon)
(More) Examples!
Example 6 - Solve for x
360
e
n
360
e
7
e  51.4
x°
Example 7 - Find the measures of
each angle of a regular dodecagon.
360
e
n
360
e
12
e  30
i  e  180
i  150
OR
(n  2)180
i
n
(12  2)180
i
12
(10)180
i
12
1800
i
12
i  150