Polygons and their Angle Measures

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Polygons and their Angle Measures
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Essential Questions

• How do I identify and classify polynomials?
• How do I find the measures of interior and
  exterior angles of polygons?
• How do I use measures of angles of polygons to
  solve problems?

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• Polygon- a plane figure that (1) is formed by 3
  or more segments, such that no two sides with a
  common endpoint are collinear and (2) each side
  intersects exactly 2 other sides , 1 at each
  endpoint.
• Means the figure is closed and no sides cross
  over each other.
• Vertex- the endpoints of each side
• Think of them as the corner points!

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Example
vertex
side
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ExampleIs the figure a polygon?
yes
yes
no
no
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• Number of sides
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 12
• n

• Type of polygon
• Triangle
• Quadrilateral
• Pentagon
• Hexagon
• Heptagon
• Octagon
• Nonagon
• Decagon
• Dodecagon
• N-gon

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Special types of Polygons

• Convex- no line that contains a side of a polygon
  goes through its interior
• Concave- opposite of a convex
• Equilateral- all sides are ?
• Equiangular- all angles are ?
• Regular polygon- equilateral and equiangular

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• Diagonal- a segment that joins 2 nonconsecutive
  vertices

A
segments AE and AD are diagonals
C
B
D
E
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Interior Angles of a Quadrilateral

• The sum of the measures of the interior ?s of a
  quad 360o.

A
B
A
1
2
m?1 m?2 m?3 m?4 360o
3
4
C
D
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ExampleFind x
(
X

• xx5555360
• 2x110360
• 2x250
• X125

55o
)
X
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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.

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Corollary to the Polygon Interior Angles Theorem

• The measure of each interior angle of a regular
  n-gon is

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Example

• Find the value of x in the diagram shown.

108146101113153x720
621 x 720
x 99
The measure of the 6th interior angle is 99!
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Try This!

• Find the value of x.

93º123º102º100ºxº 540º
418 x 540
x 122
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Example

• The measure of each interior angle of a regular
  polygon is 140. How many sides does the polygon
  have?

The polygon has 9 sides.
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Try This!

• The measure of each interior angle of a regular
  polygon is 165º. How many sides does the polygon
  have?

The polygon has 24 sides.
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Polygon Exterior Angles Theorem

• The sum of the measures of the exterior angles of
  a convex polygon, one angle at each vertex, is
  360º.

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Corollary to the Polygon Exterior Angles Theorem

• The measure of each exterior angle of a regular
  n-gon is

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Example

• Find the value of x.

2x?2x?4x?3x?x? 360?
12x 360
x 30
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Example

• Find the value of x.

x 360/7
x ? 51.43º
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Try This!

• Find the value of y.

y? y? 2y? 2y? 360?
6y 360
y 60
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Summarizer

• Explain in words how to find the measure of each
  interior angle and each exterior angle in a
  regular polygon.