1. Geometry 3-4 Polygon Angle Sum Theorems

2. Vocabulary

5. Polygons Name a polygon with consecutive vertices

6. Vocabulary Diagonal – A line segment that connects two non-consecutive vertices Convex – No diagonal goes outside the polygon Concave – At least one diagonal is outside the polygon

7. Vocabulary Equilateral – A polygon with all sides congruent Equiangular – A polygon with all angles congruent Regular Polygon – A polygon that is both equilateral and equiangular

8. Angle Exploration Equilateral Triangle 60°

9. Angle Exploration Square 90°

10. Angle Exploration Regular Pentagon 108°

11. Angle Exploration Regular Hexagon 120°

12. Angle Exploration Regular Heptagon 129°

13. Angle Exploration Regular Octagon 135°

14. Angle Exploration Regular Nonagon 140°

15. Angle Exploration Regular Decagon 144°

16. Angle Exploration Regular Dodecagon 147°

17. Angle Exploration Summarize our results SidesDegrees 360° 490° 5108° 6120° 7129° 8135° 9140° 10144° 11147° Is there a correlation between the number of sides and the degrees?

18. Angle Exploration What is the sum of the angles in each of the polygons

19. Angle Exploration Summarize our results SidesTotalDegrees 3180°60° 4360°90° 5540°108° 6720°120° 7900°129° 81080°135° 91260°140° 101440°144° 111620°147° Total Degrees in each polygon. Do you see a pattern?

20. Polygon Angle Sum Theorem The sum of the angles of the interior angles of a convex n- gon is (n-2)180°

21. Interior Angles in a Polygon What are the angle measures if the polygon is regular (n-2)180°/n

22. X=85

23. X=97

24. 8

25. 36

26. 19.571

27. 60

28. 17

29. 104

30. X=96

31. X=117

32. Geometry 3-4b Polygon Exterior Angle Sum

33. Review – Name the angle relationship 1 2 Vertical Pair

34. Review – Name the angle relationship 160° Supplementary Angles 20°

35. Review – Name the angle relationship 1 2 Linear Pair

36. Review – Name the angle relationship 55° Complementary Angles 35°

37. Review – Name the angle relationship 1 2 Vertical Pair

38. Review – Name the angle relationship 60° Complementary Angles 30°

39. Review – Name the angle relationship 1 2 Linear Pair

40. Review – Name the angle relationship 90° Supplementary Angles 90°

41. What is an interior angle? Any of the six angles inside the polygon

42. What is an exterior angle? When the sides of the polygon are extended, the angle that is adjacent to the interior angle

43. What is an exterior angle? When the sides of the polygon are extended, the angle that is adjacent to the interior angle

44. Spot the exterior angle

45. Exterior Angles Which side do you extend to create an exterior angle?

46. Exterior Angles Either one, these two angles are congruent, because they are vertical angles.

47. Exterior angle What is the value of all the exterior angles in any polygon?

48. Exterior Angle Investigation Get your supplies –Printer Paper –Straightedge –Protractor

49. Exterior Angle Investigation Draw a large polygon on your paper Pentagon or larger Extend each of its sides to form exterior angles

50. Exterior Angle Investigation Measure all of the interior angles except one. Write the value of the interior angle on the angle. Use the polygon sum conjecture to calculate the measure of the last angle before you measure it.

51. Exterior Angle Investigation Use the linear pair conjecture to calculate the value of each exterior angle. Write down the value on the angle Add up all the exterior measures

52. Polygon Exterior Angle Sum Theorem The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°

54. Exterior angle What is the measure of an exterior angle in a regular polygon?

55. Exterior Angle Theorem The measure of each exterior angle of a regular n-gon is 360°/n

56. Equiangular Polygon Conjecture You can find the measure of each interior angle of an equiangular n-gon by using either of these formulas: 180° - 360°/n (180°(n-2))/n

69. Homework Pages 147 – 150 1 – 4, 12, 13, 17, 20, 47 – 49, 54, 71, 73 - 79

70. Homework Pages 147 – 150 12, 13, 17, 20, 54, 71, 74 - 79