1. Geometry
3.5 Angles of a Polygon

2. Polygons (“many angles”)
have vertices, sides, angles, and exterior angles
are named by listing consecutive vertices in order A B C F
Hexagon ABCDEF D E

3. Polygons each segment intersects two other segments
formed by line segments, no curves
the segments enclose space
each segment intersects two other segments

4. Polygons
Not Polygons

5. Diagonal of a Polygon A segment connecting two nonconsecutive vertices
Diagonals

6. Convex Polygons No side ”collapses” in toward the center
Easy test : RUBBER BAND stretched around the figure would have the same shape…….

7. Convex Polygons
Nonconvex
Polygons

8. From now on…….
When the textbook refers to polygons, it means convex polygons

9. Polygons are classified by number of sides
Number of sides Name of Polygon
3 triangle
4 quadrilateral
5 pentagon
6 hexagon
8 octagon
10 decagon
n n-gon

10. 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)
6 sides, 4 triangles
Angle sum = 4 (180)
DO YOU SEE A PATTERN ?

11. 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)

12. Theorem
The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)180.
5 sides. 3 triangles.
Sum of angle measures is
(5-2)(180) = 3(180)
= 540
Example:

13. Exterior Angles of a Polygon3 2 2 1 4 3 5 1 4 5
Draw the exterior angles
Put them together
The sum = 360
Works with every polygon!

14. Theorem
The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.

15. Regular Polygons
If a polygon is both equilateral and equiangular it is called a regular polygon
120
120
120
120
120
120
120
120
120
120
120
120
Equilateral
Equiangular
Regular

16. Example 1 A polygon has 8 sides (octagon.) Find:
The interior angle sum
The exterior angle sum
n=8, so
(8-2)180 = 6(180)
= 1080
360

17. 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

18. Example 3 How many sides does a regular polygon have if:
the measure of each exterior angle is 45
360 = = 45n
n n = 8 8 sides: an octagon
the measure of each interior angle is 150
(n-2)180 = 150 (n-2)180 = 150n
n 180n – 360 = 150n
- 360 = - 30n
n = sides

19. 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

20. Homework
pg #1-17, skip 7, bring compass