Geometry

1. Chapter 6
Quadrilaterals
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Tuesday, April 10, 2012

2. Section 6-1
Angles of Polygons
Tuesday, April 10, 2012

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?
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4. Vocabulary
1. Diagonal:
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5. Vocabulary
1. Diagonal: A segment in a polygon that connects a vertex with
another vertex that is nonconsecutive
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6. Theorems
6.1 - Polygon Interior Angles Sum:
6.2 - Polygon Exterior Angles Sum:
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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 10, 2012

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°
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9. Polygons and Sides
Tuesday, April 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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
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16. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
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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
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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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

25. Example 2
Find the measure of each interior angle of parallelogram RSTU.
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26. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
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27. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
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28. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
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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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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.
http://www.parkcitycenter.com/directory
S =(n−2)180
Tuesday, April 10, 2012

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.
http://www.parkcitycenter.com/directory
S =(n−2)180
S =(8−2)180
Tuesday, April 10, 2012

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.
http://www.parkcitycenter.com/directory
S =(n−2)180
S =(8−2)180
=(6)180
Tuesday, April 10, 2012

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.
http://www.parkcitycenter.com/directory
S =(n−2)180
S =(8−2)180
=(6)180
=1080°
Tuesday, April 10, 2012

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.
http://www.parkcitycenter.com/directory
S =(n−2)180
S =(8−2)180
=(6)180
=1080°
1080°
8
Tuesday, April 10, 2012

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.
http://www.parkcitycenter.com/directory
S =(n−2)180
S =(8−2)180
=(6)180
=1080°
1080°
8
=135°
Tuesday, April 10, 2012

48. Example 4
The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
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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
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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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

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 10, 2012

61. Check Your Understanding
Review #1-11 on p. 393
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62. Problem Set
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63. Problem Set
p. 394 #13-37 odd, 49, 59
"They can because they think they can." - Virgil
Tuesday, April 10, 2012