Polygons and Their Polygons and Their AnglesAngles

1-6 and 6-1

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.

One name for this polygon is _________________________

Diagonal: a segment connecting two NONconsecutive vertices of a polygon.

Diagonal Example: __________________

ABCDEF

AC or AD or AE

Convex Polygons - polygons where no diagonalgoes outside the figure.

Concave Polygons - polygons where any diagonal goesoutside of the figure. Concave polygons “cave” in.

ClassifyinClassifying g

PolygonsPolygonsYou can classify a polygon by the number of sides it

has. YOU WILL BE EXPECTED TO KNOW

THESE!!

INTERIOR ANGLE SUM INTERIOR ANGLE SUM THEOREMTHEOREM

The sum of the measures of the angles in a convex polygonwith n sides is (n - 2)180

Exterior Angle Sum Exterior Angle Sum TheoremTheorem

The sum of the measure of the exterior angles of ANY convex polygon, one at each vertex is 360

ExampleExampleFind the interior and exterior angle sums for each polygon:

1. quadrilateral

2. 12-gon

3. hexagon

4. nonagon

5. decagon

6. pentagon

7. octagon

8. 18-gon

(4-2)180 = 360

Exterior Angle sum is always 360

(12-2)180 = 1800Ext. Angle Sum = 360

(6-2)180 = 720Ext. Angle Sum = 360

(9-2)180 = 1260Ext. Angle Sum = 360

(10-2)180 = 1440Ext. Angle Sum = 360

(5-2)180 = 540Ext. Angle sum = 360

(8-2)180 = 1080Ext Angle Sum = 360

(18-2)180 = 2880Ext Angle Sum = 360

Example 2Example 2Find the value of x

Since there are 5 sides, then the interior angle sum is (5-2)180 or

540.Then take 360 - 90 - 90 - 160 -

150 to get x. You should get 50 Do the same for the others. Count

the number of sides and figurethe interior angle sum. Thensubtract out the angles that

you already know.

You should get x = 100 You should get x = 90

You know there are 4 sides, so the interior angle sum is 360.

Take 360 - 60 and you get 300.Then divide by 3 and each

angle is 100.

Regular PolygonsRegular PolygonsRegular 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

Example 3Example 3For each REGULAR polygon, find the measure of

each interior angle and exterior angle.

13. triangle

14. quadrilateral

15. hexagon

16. decagon

17. 15-gon

Interior =(3-2)180 / 3 = 60Exterior = 360 / 3 = 120

Interior = (4-2)180 / 4 = 90Exterior = 360 / 4 = 90

Interior = (6-2)180 / 6 = 120Exterior = 360 / 6 = 60

Interior = (10-2)180 / 10 = 144Exterior = 360 / 10 = 36

Interior = (15-2)180 / 15 = 156Exterior = 360 / 15 = 24

NOTICE: interior and exterior angles

add to 180!!!!

Example 4Example 4How many sides does a regular polygon have

if the measure of each exterior angle is:

18. 60

19. 15

20. 120

Just take 360 divided by each angle to get your answer

6 sides

24 sides

3 sides

Example 5Example 5How many sides does a regular polygon have if

the measure of each interior angle is:

21. 60

22. 160

23. 144

Since the interior and exterior angles

Add to 180, find the exterior angle first!!!

Interior angle is 60, so exterior angle is 180-60 = 120. Now do 360 divided by

120. You should get 3.

Interior angle is 160. 180-160 = 20, so exterior angle is 20. Now do 360 divided by

20. You get 18Interior angle is 144. 180-144 = 36, so

exterior angle is 36. Now do 360 divided by 36. You get 10.

Have a great day!!Have a great day!!