3.4 The Polygon Angle-Sum Theorems

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3.4 The Polygon Angle-Sum Theorems

• Chapter 3 Parallel and Perpendicular Lines

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3.4 The Polygon Angle-Sum Theorems

• Polygon a closed plane figure with at least
  three sides that are segments

Not a polygon Not enclosed
Not a polygon Two sides intersect
A polygon
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Naming a Polygon

• Name a polygon by its vertices.

A
ABCDE or AEDCB Start at one vertex and go around
in order
B
E
C
D
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Naming a Polygon

• Three polygons are pictured. Name each polygon

L
M
P
O
N
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Classifying a Polygon by the number of sides
Sides Name
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n n-gon
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Convex vs. Concave

• A Convex Polygon has all vertices pointing out
• A Concave Polygon has one or more vertices
• caving in

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Classify

• Classify each polygon by its sides. Identify
  each as convex or concave

Hexagon Convex
Octagon Concave
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Sum of Polygon Angle Measures

• Use triangles to figure out the sum of the angles
  in each polygon

of Sides of Triangles Total Degrees
of Sides of Triangles Total Degrees
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Sum of Polygon Angle Measures
Number of Sides Number of Triangles Total Degrees inside Polygon
3 1 180
4
5
6
n
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Theorem 3-9 Polygon Angle Sum Theorem

• The sum of the measures of the angles in a
  polygon
• is (n 2)180.

Find the sum of the measure of the angles of a
15-gon.
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Polygon Angle Sum

• The sum of the measures of the angles of a given
  polygon is 720. How many sides does the polygon
  have?

Use (n 2)180
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Using Polygon Angle-Sum Theorem

• Find the measure of ltY in pentagon TVYMR at the
  right.

R
T
Use (n 2)180
135
M
90
Y
V
Write an equation to solve for ltY
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Using Polygon Angle-Sum Theorem

• Pentagon ABCDE has 5 congruent angles. Find the
  measure of each angle.

Use the Polygon Angle-Sum Theorem (n 2)180
Divide the total number of degrees by the number
of angles
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Exterior Angles

• What do you notice about each set of exterior
  angles?

80
75
115
2
1
150
99
130
71
70
88
1
86
3
2
3
70
46
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Theorem 3-10 Polygon Angle-Sum Theorem

• The sum of one set of exterior angles for any
  polygon is 360.

1
5
2
4
3
mlt1 mlt2 mlt3 mlt4 mlt5 360
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Polygons

• Equilateral Polygon all sides congruent
• Equiangular Polygon all angles congruent
• Regular Polygon all sides and all angles
  congruent (equiangular and equilateral)
• If a polygon is a regular polygon then all of
  the exterior angles are also congruent.

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Homework

• Pg 147 1-25, 40-44, 47-49