1. Angles of Polygons

2. Polygons can be CONCAVE or CONVEX

3. Concave and Convex Polygons
If a polygon has an indentation (or cave), the polygon is called a concave polygon. Any polygon that does not have an indentation is called a convex polygon.
Any two points in the interior of a convex polygon can be connected by a line segment that does not cut or cross a side of the polygon.
Concave polygon
Convex polygon
We will only be discussing CONCAVE polygons

4. Triangle Octagon Quadrilateral Nonagon Pentagon Decagon Dodecagon
Hexagon
n-gon
Heptagon
Hendecagon

5. Important Terms A VERTEX is the point of intersection of two sides F A
CONSECUTIVE VERTICES are two endpoints of any side. F A B C D E
A segment whose endpoints are two
nonconsecutive vertices is called a DIAGONAL.
Sides that share a vertex are called CONSECUTIVE SIDES.

7. Tear off two vertices….

8. Line up the 3 angles (all vertices touching)

10. A straight line = 180°

11. ALWAYS!!! Angle sum of a Triangle 180° <1 + <2 + <3 = 180° 2

12. Consider a Quadrilateral
What is the angle sum?
<1 + <2 + <3 + <4 = ?

13. Quadrilateral Draw a diagonal…what do you get? Two triangles 2 3 5 1 46

14. Quadrilateral 180° 180° Each triangle = 180°2 3 5
180°
180° 1 4
Therefore the two triangles together = 360° 6

15. Angle sum of a Quadrilateral
180° + 180° =
360°

16. Consider a Pentagon
What is the angle sum?

17. Pentagon
Draw the diagonals from 1 vertex
How many triangles?

18. Angle sum of a Pentagon 180° 180° 180°
Draw the diagonals from 1 vertex
180°
180°
180°

19. Continue this process through Decagon
Draw the diagonals from 1 vertex

20. Continue this process through Decagon
Draw the diagonals from 1 vertex

21. What about a 52-gon? What is the angle sum? Can you find the pattern?1
180° 2
360° 3
540° 4
720° 5
900° 6
1080°

22. Find the nth term 7
1260° 8
1440°
n - 2
(n – 2)(180)

23. pentagon 5(20) - 5 m1 = = 95 Find m1. (4x + 15) 2 (5x - 5)3
110
(5x - 5)
(4x + 15)
(8x - 10)
m1 =
5(20) - 5
= 95
540
17x + 200= 540
17x = 340
5x x x =
x = 20

24. More important terms Interior Angles
Exterior Angles
the SUM of an interior angle and it’s corresponding exterior angle = 180o

25. Sums of Exterior Angles1 2 3 4 5 6
180
180
180•3 = 540
180
Sum of Interior & Exterior Angles =
540
Sum of Interior Angles =
180
Sum of Exterior Angles =
540- 180=
360

26. Sums of Exterior Angles
180
180
180
180•4 = 720
180
Sum of Interior & Exterior Angles =
720
Sum of Interior Angles =
360
Sum of Exterior Angles =
720- 360=
360

27. Sums of Exterior Angles
Sum of Interior & Exterior Angles =
180•5 =
900
Sum of Interior Angles =
180•3 =
540
Sum of Exterior Angles =
360
900- 540=

28. What conclusion can you come up with regarding the exterior angle sum of a CONVEX n-polygon??
Sum of Interior & Exterior Angles =
180n
Sum of Interior Angles =
180(n-2) = 180n - 360
Sum of Exterior Angles =
180n – (180n – 360)

29. The exterior angle sum of a CONVEX polygon =
360°

30. Important Terms EQUILATERAL - All sides are congruent
EQUIANGULAR - All angles are congruent
REGULAR - All sides and angles are congruent

31. Interior Angle Measure of a REGULAR polygons
60°
90°
Equilateral Triangle
Angle measure = 60°
Square
Angle measure = 90°
These are measurement that we generally know at this time,
But what about the other regular polygons?
How do we calculate the interior angle measure?

32. Pentagon
72°
108°
72°
108°
108°
72°
72°
72°
108°
108°

33. Interior Angle Measure of a REGULAR polygons
108°
120°
Calculate by:
Angle Sum
Number of sides
135°