1. Polygons and Their Angles

2. Polygon: A closed figure
Polygon: A closed figure. They have vertices, sides, angles, and exterior angles.
You name a polygon by just listing its vertices in order around the polygon.
ABCDEF
One name for this polygon is _________________________
Diagonal: a segment connecting two NONconsecutive vertices of a polygon.
AC or AD or AE
Diagonal Example: __________________

3. Convex Polygons - polygons where no diagonal goes outside the figure.
Concave Polygons - polygons where any diagonal goes
outside of the figure. Concave polygons “cave” in.

4. Classifying Polygons
You can classify a polygon by the number of sides it has. YOU WILL BE EXPECTED TO KNOW THESE!!

5. INTERIOR ANGLE SUM THEOREM
The sum of the measures of the angles in a convex polygon
with n sides is (n - 2)180

6. Exterior Angle Sum Theorem
The sum of the measure of the exterior angles of ANY
convex polygon, one at each vertex is 360

7. Example Find the interior and exterior angle sums for each polygon:
1. quadrilateral
gon
3. hexagon
4. nonagon
(4-2)180 = 360
5. decagon
6. pentagon
7. octagon
8. 18-gon
(10-2)180 = 1440
Ext. Angle Sum = 360
Exterior Angle sum is always 360
(12-2)180 = 1800
Ext. Angle Sum = 360
(5-2)180 = 540
Ext. Angle sum = 360
(6-2)180 = 720
Ext. Angle Sum = 360
(8-2)180 = 1080
Ext Angle Sum = 360
(9-2)180 = 1260
Ext. Angle Sum = 360
(18-2)180 = 2880
Ext Angle Sum = 360

8. Example 2 Find the value of x You should get x = 100
Since there are 5 sides, then the interior angle sum is (5-2)180 or 540.
Then take to get x.
You should get 50
Do the same for the others. Count
the number of sides and figure
the interior angle sum. Then
subtract out the angles that
you already know.
You know there are 4 sides, so
the interior angle sum is 360.
Take and you get 300.
Then divide by 3 and each
angle is 100.

9. Regular Polygons Regular Polygons - a polygon that is BOTH equilateral
AND equiangular
If you see the word REGULAR, it means the figure is special
and you can divide by the number of sides to get
individual angle measures

10. NOTICE: interior and exterior angles add to 180!!!!
Example 3
For each REGULAR polygon, find the measure of
each interior angle and exterior angle.
13. triangle
14. quadrilateral
15. hexagon
Interior =(3-2)180 / 3 = 60
16. decagon
gon
Exterior = 360 / 3 = 120
Interior = (10-2)180 / 10 = 144
Exterior = 360 / 10 = 36
Interior = (4-2)180 / 4 = 90
Exterior = 360 / 4 = 90
Interior = (15-2)180 / 15 = 156
Exterior = 360 / 15 = 24
Interior = (6-2)180 / 6 = 120
Exterior = 360 / 6 = 60
NOTICE: interior and exterior angles add to 180!!!!

11. Example 4 How many sides does a regular polygon have
if the measure of each exterior angle is:
Just take 360 divided by each angle to get your answer
6 sides
24 sides
3 sides

12. Example 5 How many sides does a regular polygon have if
the measure of each interior angle is:
Since the interior and exterior angles
Add to 180, find the exterior angle first!!!
Interior angle is 60, so exterior angle is
= 120. Now do 360 divided by 120. You should get 3.
Interior angle is = 20, so exterior angle is 20. Now do 360 divided by 20. You get 18
Interior angle is = 36, so exterior angle is 36. Now do 360 divided by 36. You get 10.

13. Have a great day!!