Geometry Section 6-1 1112

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Angles of Polygon

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Geometry Section 6-1 1112

  1. 1. Chapter 6 Quadrilaterals Created at wordle.net Tuesday, April 29, 14
  2. 2. Section 6-1 Angles of Polygons Tuesday, April 29, 14
  3. 3. Essential Questions How do you find and use the sum of the measures of the interior angles of a polygon? How do you find and use the sum of the measures of the exterior angles of a polygon? Tuesday, April 29, 14
  4. 4. Vocabulary 1. Diagonal: Tuesday, April 29, 14
  5. 5. Vocabulary 1. Diagonal: A segment in a polygon that connects a vertex with another vertex that is nonconsecutive Tuesday, April 29, 14
  6. 6. Theorems 6.1 - Polygon Interior Angles Sum: 6.2 - Polygon Exterior Angles Sum: Tuesday, April 29, 14
  7. 7. Theorems 6.1 - Polygon Interior Angles Sum: The sum of the interior angle measures of a convex polygon with n sides is found with the formula S =(n−2)180 6.2 - Polygon Exterior Angles Sum: Tuesday, April 29, 14
  8. 8. Theorems 6.1 - Polygon Interior Angles Sum: The sum of the interior angle measures of a convex polygon with n sides is found with the formula S =(n−2)180 6.2 - Polygon Exterior Angles Sum: The sum of the exterior angle measures of a convex polygon, one at each vertex, is 360° Tuesday, April 29, 14
  9. 9. Polygons and Sides Tuesday, April 29, 14
  10. 10. Polygons and Sides Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons # sides 3 4 5 6 7 Name Triangle Quadrilateral Pentagon Hexagon Heptagon Tuesday, April 29, 14
  11. 11. Polygons and Sides Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons # sides 3 4 5 6 7 Name Triangle Quadrilateral Pentagon Hexagon Heptagon Tuesday, April 29, 14
  12. 12. Polygons and Sides Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons # sides 3 4 5 6 7 Name Triangle Quadrilateral Pentagon Hexagon Heptagon Tuesday, April 29, 14
  13. 13. Polygons and Sides Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons # sides 3 4 5 6 7 Name Triangle Quadrilateral Pentagon Hexagon Heptagon Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons # sides 8 9 10 12 n Name Octagon Nonagon Decagon Dodecagon n-gon Tuesday, April 29, 14
  14. 14. Polygons and Sides Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons # sides 3 4 5 6 7 Name Triangle Quadrilateral Pentagon Hexagon Heptagon Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons # sides 8 9 10 12 n Name Octagon Nonagon Decagon Dodecagon n-gon Tuesday, April 29, 14
  15. 15. Polygons and Sides Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons # sides 3 4 5 6 7 Name Triangle Quadrilateral Pentagon Hexagon Heptagon Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons # sides 8 9 10 12 n Name Octagon Nonagon Decagon Dodecagon n-gon Tuesday, April 29, 14
  16. 16. Example 1 Find the sum of the measures of the interior angles of the following. a. Nonagon b. 17-gon Tuesday, April 29, 14
  17. 17. Example 1 Find the sum of the measures of the interior angles of the following. a. Nonagon b. 17-gon S =(n−2)180 Tuesday, April 29, 14
  18. 18. Example 1 Find the sum of the measures of the interior angles of the following. a. Nonagon b. 17-gon S =(n−2)180 S =(9−2)180 Tuesday, April 29, 14
  19. 19. Example 1 Find the sum of the measures of the interior angles of the following. a. Nonagon b. 17-gon S =(n−2)180 S =(9−2)180 =(7)180 Tuesday, April 29, 14
  20. 20. Example 1 Find the sum of the measures of the interior angles of the following. a. Nonagon b. 17-gon S =(n−2)180 S =(9−2)180 =(7)180 =1260° Tuesday, April 29, 14
  21. 21. Example 1 Find the sum of the measures of the interior angles of the following. a. Nonagon b. 17-gon S =(n−2)180 S =(9−2)180 =(7)180 =1260° S =(n−2)180 Tuesday, April 29, 14
  22. 22. Example 1 Find the sum of the measures of the interior angles of the following. a. Nonagon b. 17-gon S =(n−2)180 S =(9−2)180 =(7)180 =1260° S =(n−2)180 S =(17−2)180 Tuesday, April 29, 14
  23. 23. Example 1 Find the sum of the measures of the interior angles of the following. a. Nonagon b. 17-gon S =(n−2)180 S =(9−2)180 =(7)180 =1260° S =(n−2)180 S =(17−2)180 =(15)180 Tuesday, April 29, 14
  24. 24. Example 1 Find the sum of the measures of the interior angles of the following. a. Nonagon b. 17-gon S =(n−2)180 S =(9−2)180 =(7)180 =1260° S =(n−2)180 S =(17−2)180 =(15)180 = 2700° Tuesday, April 29, 14
  25. 25. Example 2 Find the measure of each interior angle of parallelogram RSTU. Tuesday, April 29, 14
  26. 26. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 Tuesday, April 29, 14
  27. 27. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 Tuesday, April 29, 14
  28. 28. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 Tuesday, April 29, 14
  29. 29. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° Tuesday, April 29, 14
  30. 30. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 Tuesday, April 29, 14
  31. 31. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 Tuesday, April 29, 14
  32. 32. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 −8 −8 Tuesday, April 29, 14
  33. 33. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 −8 −8 32x =352 Tuesday, April 29, 14
  34. 34. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 −8 −8 32x =352 32 32 Tuesday, April 29, 14
  35. 35. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 −8 −8 32x =352 32 32 x =11 Tuesday, April 29, 14
  36. 36. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 −8 −8 32x =352 32 32 x =11 m∠R = m∠T =5(11) Tuesday, April 29, 14
  37. 37. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 −8 −8 32x =352 32 32 x =11 m∠R = m∠T =5(11) =55° Tuesday, April 29, 14
  38. 38. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 −8 −8 32x =352 32 32 x =11 m∠R = m∠T =5(11) =55° m∠S = m∠U =11(11)+4 Tuesday, April 29, 14
  39. 39. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 −8 −8 32x =352 32 32 x =11 m∠R = m∠T =5(11) =55° m∠S = m∠U =11(11)+4 =121+4 Tuesday, April 29, 14
  40. 40. Example 2 Find the measure of each interior angle of parallelogram RSTU. S =(n−2)180 S =(4−2)180 =(2)180 =360° 11x +4+5x +11x +4+5x =360 32x +8 =360 −8 −8 32x =352 32 32 x =11 m∠R = m∠T =5(11) =55° m∠S = m∠U =11(11)+4 =121+4 =125° Tuesday, April 29, 14
  41. 41. Example 3 Park City Mall is designed so that eight walkways meet in a central area in the shape of a regular octagon. Find the measure of one of the interior angles of the octagon. http://www.parkcitycenter.com/directory Tuesday, April 29, 14
  42. 42. Example 3 Park City Mall is designed so that eight walkways meet in a central area in the shape of a regular octagon. Find the measure of one of the interior angles of the octagon. S =(n−2)180 http://www.parkcitycenter.com/directory Tuesday, April 29, 14
  43. 43. Example 3 Park City Mall is designed so that eight walkways meet in a central area in the shape of a regular octagon. Find the measure of one of the interior angles of the octagon. S =(n−2)180 S =(8−2)180 http://www.parkcitycenter.com/directory Tuesday, April 29, 14
  44. 44. Example 3 Park City Mall is designed so that eight walkways meet in a central area in the shape of a regular octagon. Find the measure of one of the interior angles of the octagon. S =(n−2)180 S =(8−2)180 =(6)180 http://www.parkcitycenter.com/directory Tuesday, April 29, 14
  45. 45. Example 3 Park City Mall is designed so that eight walkways meet in a central area in the shape of a regular octagon. Find the measure of one of the interior angles of the octagon. S =(n−2)180 S =(8−2)180 =(6)180 =1080° http://www.parkcitycenter.com/directory Tuesday, April 29, 14
  46. 46. Example 3 Park City Mall is designed so that eight walkways meet in a central area in the shape of a regular octagon. Find the measure of one of the interior angles of the octagon. S =(n−2)180 S =(8−2)180 =(6)180 =1080° 1080° 8 http://www.parkcitycenter.com/directory Tuesday, April 29, 14
  47. 47. Example 3 Park City Mall is designed so that eight walkways meet in a central area in the shape of a regular octagon. Find the measure of one of the interior angles of the octagon. S =(n−2)180 S =(8−2)180 =(6)180 =1080° 1080° 8 =135° http://www.parkcitycenter.com/directory Tuesday, April 29, 14
  48. 48. Example 4 The measure of an interior angle of a regular polygon is 150°. Find the number of sides in the polygon. Tuesday, April 29, 14
  49. 49. Example 4 The measure of an interior angle of a regular polygon is 150°. Find the number of sides in the polygon. 150n =(n−2)180 Tuesday, April 29, 14
  50. 50. Example 4 The measure of an interior angle of a regular polygon is 150°. Find the number of sides in the polygon. 150n =(n−2)180 150n =180n−360 Tuesday, April 29, 14
  51. 51. Example 4 The measure of an interior angle of a regular polygon is 150°. Find the number of sides in the polygon. 150n =(n−2)180 150n =180n−360 −180n −180n Tuesday, April 29, 14
  52. 52. Example 4 The measure of an interior angle of a regular polygon is 150°. Find the number of sides in the polygon. 150n =(n−2)180 150n =180n−360 −180n −180n −30n = −360 Tuesday, April 29, 14
  53. 53. Example 4 The measure of an interior angle of a regular polygon is 150°. Find the number of sides in the polygon. 150n =(n−2)180 150n =180n−360 −180n −180n −30n = −360 −30 −30 Tuesday, April 29, 14
  54. 54. Example 4 The measure of an interior angle of a regular polygon is 150°. Find the number of sides in the polygon. 150n =(n−2)180 150n =180n−360 −180n −180n −30n = −360 −30 −30 n =12 Tuesday, April 29, 14
  55. 55. Example 4 The measure of an interior angle of a regular polygon is 150°. Find the number of sides in the polygon. 150n =(n−2)180 150n =180n−360 −180n −180n −30n = −360 −30 −30 n =12 There are 12 sides to the polygon Tuesday, April 29, 14
  56. 56. Example 5 Find the value of x in the diagram. m∠1=5x +5, m∠2 =5x, m∠3= 4x −6, m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12, m∠7 = 2x +3 Tuesday, April 29, 14
  57. 57. Example 5 Find the value of x in the diagram. m∠1=5x +5, m∠2 =5x, m∠3= 4x −6, m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12, m∠7 = 2x +3 5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360 Tuesday, April 29, 14
  58. 58. Example 5 Find the value of x in the diagram. m∠1=5x +5, m∠2 =5x, m∠3= 4x −6, m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12, m∠7 = 2x +3 5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360 31x −12 =360 Tuesday, April 29, 14
  59. 59. Example 5 Find the value of x in the diagram. m∠1=5x +5, m∠2 =5x, m∠3= 4x −6, m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12, m∠7 = 2x +3 5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360 31x −12 =360 31x =372 Tuesday, April 29, 14
  60. 60. Example 5 Find the value of x in the diagram. m∠1=5x +5, m∠2 =5x, m∠3= 4x −6, m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12, m∠7 = 2x +3 5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360 31x −12 =360 31x =372 x =12 Tuesday, April 29, 14
  61. 61. Problem Set Tuesday, April 29, 14
  62. 62. Problem Set p. 394 #1-37 odd, 49, 59 "They can because they think they can." - Virgil Tuesday, April 29, 14

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