Angles of a Polygon

1
Angles of a Polygon

• Lesson 3.5
• Pre-AP Geometry

2
Objectives

• Recognize and name convex polygons and regular
  polygons.
• Find the measures of interior angles and exterior
  angles of convex polygons.

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Definition Polygon

• A closed geometric figure in a plane formed by
  connecting line segments endpoint to endpoint
  with each segment intersecting exactly two others.

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Definition Convex Polygon

• A convex polygon is a simple polygon whose
  interior is a convex set. The following
  properties of a simple polygon are all equivalent
  to convexity
• Every internal angle is less than 180 degrees
  or equal to 180 degrees.
• Every line segment between two vertices of the
  polygon does not go exterior to the polygon
  (i.e., it remains inside or on the boundary of
  the polygon).

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Classifying Polygons
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Classifying Polygons

• Polygons are also classified according to the
  number of sides.

A triangle is a three-sided polygon. A
quadrilateral is a four-sided polygon. A
pentagon is a five-sided polygon. A hexagon is a
six-sided polygon. A heptagon is a seven-sided
polygon. An octagon is an eight-sided polygon.
A nonagon is a nine-sided polygon. A decagon is
a ten-sided polygon.
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Classifying Polygons

• Regular Polygons
• When a polygon is both equilateral and
  equiangular, it is referred to as a regular
  polygon. For a polygon to be regular, it must
  also be convex.

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Definition Diagonal

• A diagonal of a polygon is any segment that
  joins two nonconsecutive vertices.

Segments QS , SU , UR , RT and QT are the
diagonals in this polygon.
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Angle Measures in a Polygon

• Let n the number of sides of the polygon
• then,
• the polygon has n vertices
• the polygon has n pairs of supplementary angles
• the sum of the measures of all the angles 180n

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Angle Measures in a Polygon

• If n the number of sides of the polygon, then
• 180n (interior angle sum) (exterior angle
  sum)
• 180n 180(n-2) (exterior angle sum)
• 180n 180n 360 (exterior angle sum)
• 360 exterior angle sum.

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Theorem 3-13

• The sum of the measures of the angles of a
  convex polygon with n sides is (n - 2)180.

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Theorem 3-14

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

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Review Quiz

• Find the measure of an interior angle
• For a regular quadrilateral.
• For a regular octagon.
• For a regular polygon with 15 sides.
• For a regular polygon with 20 sides.

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Review Quiz - Solutions

• Find the measure of an interior angle
• For a regular quadrilateral.
  90
• For a regular octagon.
  135
• For a regular polygon with 15 sides. 156
• For a regular polygon with 20 sides. 162

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Written Exercises

• Problem Set 3.5, p. 104 6 8, 9 13 (odd),
  16, 18, 20, 21, 24 - 28