Sum of Interior and Exterior Angles in Polygons

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Sum of Interior and Exterior Angles in Polygons

• Essential Question
• How can I find angle measures in polygons without
  using a protractor?
• Key Standard MM1G3a

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Polygons

• A polygon is a closed figure formed by a finite
  number of segments such that
• 1. the sides that have a common endpoint
  are noncollinear, and
• 2. each side intersects exactly two other
  sides, but only at their endpoints.

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Nonexamples
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Polygons

• Can be concave or convex.
• Concave Convex

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Polygons are named by number of sides
Number of Sides Polygon










Triangle
3
4
Quadrilateral
Pentagon
5
Hexagon
6
Heptagon
7
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
n-gon
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Regular Polygon

• A convex polygon in which all the sides are
  congruent and all the angles are congruent is
  called a regular polygon.

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• Draw a
• ? Quadrilateral ? Pentagon
• ? Hexagon ? Heptagon
• ? Octogon
• Then draw diagonals to create triangles.
• A diagonal is a segment connecting two
  nonadjacent vertices (dont let segments cross)
• Add up the angles in all of the triangles in the
  figure to determine the sum of the angles in the
  polygon.
• Complete this table

Polygon of sides of triangles Sum of interior angles
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Polygon of sides of triangles Sum of interior angles
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
n-gon
3
1
180
4
2
2 180 360
5
3
3 180 540
4
4 180 720
6
7
5
5 180 900
8
6
6 180 1080
n
n - 2
(n 2) 180
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Polygon Interior Angles Theorem

• The sum of the measures of the interior angles of
  a convex n-gon is (n 2) 180.
• Examples
• Find the sum of the measures of the interior
  angles of a 16gon.
• If the sum of the measures of the interior angles
  of a convex polygon is 3600, how many sides
  does the polygon have.
• Solve for x.

(16 2)180
2520

(n 2)180 3600
180n 3960 180 180
n 22 sides
180n 360 3600 360 360
(4 2)180 360
4x - 2
108
108 82 4x 2 2x 10 360
6x 162 6 6
2x 10
82
6x 198 360
x 27
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• Draw a quadrilateral and extend the sides.

• There are two sets of angles formed when the
  sides of a polygon are extended.
• The original angles are called interior angles.
  
• The angles that are adjacent to the
• interior angles are called exterior angles.
• These exterior angles can be formed when any side
  is extended.
• What do you notice about the interior angle and
  the exterior angle?
• What is the measure of a line?
• What is the sum of an interior angle with the
  exterior angle?

They form a line.
180
180
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If you started at Point A, and followed along the
sides of the quadrilateral making the exterior
turns that are marked, what would happen? You end
up back where you started or you would make a
circle. What is the measure of the degrees in a
circle?
A
D
B
C
360
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Polygon Exterior Angles Theorem

• The sum of the measures of the exterior angles of
  a convex polygon, one at each vertex, is 360.
• Each exterior angle of a regular polygon is 360
• 
  n
• where n is the number of sides in the
  polygon

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Example
Find the value for x.
Sum of exterior angles is 360 (4x 12) 60
(3x 13) 65 54 68 360
7x 248 360
248 248
7x
112
7 7
x 12
(4x 12)
68
60
54
(3x 13)
65
What is the sum of the exterior angles in an
octagon? What is the measure of each exterior
angle in a regular octagon?
360
360/8
45
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Classwork/Homework

• Textbook Read and study p298-299
• Complete p300-301 (1-21)
• Show your work!