Geometry
3.5 Angles of a Polygon
Polygons (“many angles”)
•
have vertices, sides, angles, and exterior angles
• are named by listing consecutive vertices in
order
A
B
C
F
E
D
Hexagon ABCDEF
Polygons
• formed
• the
by line segments, no curves
segments enclose space
• each segment intersects two other segments
Polygons
Not Polygons
Diagonal of a Polygon
A segment connecting two nonconsecutive vertices
Diagonals
Convex Polygons
No side ”collapses” in toward the center
Easy test : RUBBER BAND stretched
around the figure would have the same
shape…….
Convex
Polygons
Nonconvex
Polygons
From now on…….
When the textbook refers to
polygons, it means convex polygons
Polygons are classified by
number of sides
Number of sides
3
4
5
6
8
10
n
Name of Polygon
triangle
quadrilateral
pentagon
hexagon
octagon
decagon
n-gon
Interior Angles of a Polygon
•To find the sum of angle measures, divide
the polygon into triangles
•Draw diagonals from just one vertex
4 sides, 2 triangles
Angle sum = 2 (180)
5 sides, 3 triangles
Angle sum = 3 (180)
DO YOU SEE A PATTERN ?
6 sides, 4 triangles
Angle sum = 4 (180)
Interior Angles of a Polygon
4 sides, 2 triangles
Angle sum = 2 (180)
5 sides, 3 triangles
Angle sum = 3 (180)
6 sides, 4 triangles
Angle sum = 4 (180)
The pattern is:
ANGLE SUM = (Number of sides – 2) (180)
Theorem
The sum of the measures of the
interior angles of a convex polygon
with n sides is (n-2)180.
Example:
5 sides. 3 triangles.
Sum of angle measures is
(5-2)(180) = 3(180)
= 540
Exterior Angles of a Polygon
3
2
4
2
3
5
1
Draw the exterior angles
4
1
5
Put them together
The sum = 360
Works with every polygon!
Theorem
The sum of the measures of the
exterior angles of any convex
polygon, one angle at each vertex, is
360.
Regular Polygons
If a polygon is both equilateral and
equiangular it is called a regular polygon
120
120
120
120
120
Equilateral
120 120
120
Equiangular
120
120
120
120
Regular
Example 1
A polygon has 8 sides (octagon.) Find:
a) The interior angle sum
b) The exterior angle sum
n=8, so
(8-2)180 = 6(180)
= 1080
360
Example 2
Find the measure of each interior and exterior angle of
a regular pentagon
Interior:
(5-2)180 = 3(180) = 540
540 = 108 each
5
Exterior:
360 = 72 each
5
Example 3
How many sides does a regular polygon have if:
a)
the measure of each exterior angle is 45
360 = 45
n
b)
360 = 45n
n=8
8 sides: an octagon
the measure of each interior angle is 150
(n-2)180 = 150
n
(n-2)180 = 150n
180n – 360 = 150n
- 360 = - 30n
n = 12
12 sides
In summary…
Sum of interior angles
(n-2)180
Sum of ext. angles
360
One ext. angle
360/n
One int. angle
[(n – 2)180]/n OR supp. to 360/n
# of sides given an ext. angle
360/measue of ext. angle
# of sides given an int. angle
find the ext angle(supp to int. angle)
360/measure of ext. angle
Homework
pg. 104 #1-17, skip 7, bring compass