Welcome to Interactive Chalkboard

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8.1 Angles of Polygons
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Objectives

• Find the sum of the measures of the interior
  angles of a polygon
• Find the sum of the measures of the exterior
  angles of a polygon

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Sum of the Measures of the Interior Angles of a
Polygon

• We have already learned the name of a polygon
  depends on the number of sides in the polygon
  triangle, quadrilateral, pentagon, hexagon, and
  so forth.
• The sum of the measures of the interior angles of
  a polygon also depends on the number of sides.

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Sum of the Measures of the Interior Angles of a
Polygon

• From a previous lesson we learned the sum of the
  measures of the interior angles of a
  quadrilateral are known by dividing the
  quadrilateral into two triangles.
• You can use this triangle method to find the sum
  of the measures of the interior angles of any
  convex polygon with n sides, called an n - gon.
  

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Sum of the Measures of the Interior Angles of a
Polygon
Polygon of sides of triangles Sum of measures of interior ?s
Triangle 3 1 1 ? 180? 180?
Quadrilateral 2 ? 180? 360?
Pentagon
Hexagon
Nonagon (9)
n - gon n
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Sum of the Measures of the Interior Angles of a
Polygon

• From the previous slide, we have discovered that
  the sum of the measures of the interior angles of
  a convex n - gon is
• (n 2) ? 180?
• This relationship can be used to find the measure
  of each interior angle in a regular n - gon
  because the angles are all congruent.

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Interior Angle Sum Theorem

• Interior Angle Sum Theorem If a convex polygon
  has n sides and S is the sum of its interior
  angles, then S 180(n 2).

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Example 1
ARCHITECTURE A mall is designed so that five
walkways meet at a food court that is in the
shape of a regular pentagon. Find the sum of
measures of the interior angles of the pentagon.
Since a pentagon is a convex polygon, we can use
the Angle Sum Theorem.
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Example 1
Interior Angle Sum Theorem
Simplify.
Answer The sum of the measures of the angles is
540.
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Your Turn
A decorative window is designed to have the shape
of a regular octagon. Find the sum of the
measures of the interior angles of the octagon.
Answer 1080
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Example 2
The measure of an interior angle of a regular
polygon is 135. Find the number of sides in the
polygon.
Use the Interior Angle Sum Theorem to write an
equation to solve for n, the number of sides.
Interior Angle Sum Theorem
Distributive Property
Subtract 135n from each side.
Add 360 to each side.
Divide each side by 45.
Answer The polygon has 8 sides.
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Your Turn
The measure of an interior angle of a regular
polygon is 144. Find the number of sides in the
polygon.
Answer The polygon has 10 sides.
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Example 3
Find the measure of each interior angle.
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Example 3
Sum of measures of angles
Substitution
Combine like terms.
Subtract 8 from each side.
Divide each side by 32.
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Example 3
Use the value of x to find the measure of each
angle.
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Your Turn
Find the measure of each interior angle.
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Sum of the Measures of the Exterior Angles of a
Polygon
Interestingly, the measures of the exterior
angles of a polygon is an even easier formula.
Lets look at the following example to understand
it.
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Exterior Angle Sum Theorem

• Exterior Angle Sum TheoremIf a polygon is
  convex, then the sum of the measures of the
  exterior angles, one at each vertex, is 360.

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Example 4
Find the measures of an exterior angle and an
interior angle of convex regular nonagon
ABCDEFGHJ.
At each vertex, extend a side to form one
exterior angle.
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Example 4
The sum of the measures of the exterior angles is
360. A convex regular nonagon has 9 congruent
exterior angles.
Divide each side by 9.
Answer The measure of each exterior angle is 40.
Since each exterior angle and its corresponding
interior angle form a linear pair, the measure of
the interior angle is 180 40 or 140.
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Your Turn
Find the measures of an exterior angle and an
interior angle of convex regular hexagon ABCDEF.
Answer 60 120
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Assignment

• Pre-AP Geometry Pg. 407 14 - 40
• Geometry Pg. 407 4 15, 21 24,
  27 28, 32, 35, 36