1. Convex vs. Concave Polygons Interior Angles of Polygons Exterior Angles of Polygons Polygons

2. To be or not to be…  Polygons consist of entirely segments  Consecutive sides can only intersect at endpoints. Nonconsecutive sides do not intersect.  Vertices must only belong to one angle  Consecutive sides must be noncollinear.

3. A rose by any other name…  To name a polygon, start at a vertex and either go clockwise or counterclockwise. ab c d e f

4. Diagonals  A diagonal of a polygon is any segment that connects two nonconsecutive (nonadjacent) vertices of the polygon.

5. Convex polygons A polygon in which each interior angle has a measure less than 180.

6. Polygons can be CONCAVE or CONVEX CONVEX CONCAVE

7. Classify each polygon as convex or concave.

8. Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon 15 sides Pentadecagon

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

10. # of sides # of triangles Sum of measures of interior angles 31 1(180) = 180 42 2(180) = 360 5 33(180) = 540 644(180) = 720 n n-2 (n-2)  180

11. Regular Polygons No. of sidesNameAngle SumInterior Angle 3triangle 4 5 6 7 8 9 10 quadrilateral 180°60° 360°90° pentagon540°108° hexagon720°120° heptagon 900° 128 7/9° octagon1080°135° nonagon1260° 140° decagon1440°144°

12. If a convex polygon has n sides, then the sum of the measure of the interior angles is (n – 2)(180°)

13. Use the regular pentagon to answer the questions. A)Find the sum of the measures of the interior angles. B)Find the measure of ONE interior angle 540° 108°

14. Exterior angles of a triangle The exterior angle of a triangle is equal to the sum of the interior opposite angles. interior opposite angles exterior angle A B C D i.e.  ACD =  ABC +  BAC

15. 20° C A B D E Find  CED = 40° 40°  CDE = 40° 40°  EAB 60° = 120° 120° 55°  CAE = 85° 85°  ACE 35° = 35°  ABE = 20° 20°  AEB = 120° 120° Example

16. Two more important terms Exterior Angles Interior Angles

17. b Exterior angles of a polygon Exterior angles of a polygon add to 360°. At each vertex:interior angle + exterior angle = 180° a c e a + b + c + d + e = 360° d

18. In any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°. 1 2 3 4 5

19. 1 3 2

20. 1 3 2 4

21. Find the measure of ONE exterior angle of a regular hexagon. 60°

22. Find the measure of ONE exterior angle of a regular heptagon. 51.4°

23. Each exterior angle of a polygon is 18 . How many sides does it have? n = 20

24. The sum of the measures of five interior angles of a hexagon is 535 o. What is the measure of the sixth angle? 185°

25. x + 3x + 5x + 3x = 360 o 12x = 360 o x = 30 o Use substitution to solve for each angle measure. The measure of the exterior angle of a quadrilateral are x, 3x, 5x, and 3x. Find the measure of each angle. 30°, 90°, 150°, and 90°

26. If each interior angle of a regular polygon is 150 , then how many sides does the polygon have? n = 12

27. Find  ABC = 120° 120° Example  ADC = 60° 60°  BAC = 30° 30°  CAD = 30° 30° ABCDE is a regular hexagon with centre O. C A B D E F O  ACD  ODE  EOD = 90° = 60° 60° = 60° 60°