Section 3-4 Polygon Angle-Sum Theorem SPI 32A: Identify properties of plane figures from information given in a diagram

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Section 3-4 Polygon Angle-Sum
Theorem SPI 32A
Identify properties of plane figures from
information given in a diagram

• Objectives
• Classify Polygons
• Find the sums of the measures of the interior
  and exterior angle of polygons

• Polygon
• closed plane figure with at least 3 sides that
  are segments
• the sides intersect only at their endpoints
• no adjacent sides are collinear

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Classify Polygons
Name Polygons By Their
Vertices Start at any vertex and list the
vertices consecutively in a clockwise direction
(ABCDE or CDEAB, etc)
Sides Name by line segment naming convention
Angles Name by angle naming convention
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Classify Polygons by the Number of Sides
Most Common Polygons
Number of Sides Name









3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
n n-gon
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Classify Polygons as Convex or Concave
Convex Polygon Has no diagonals with points
outside the polygon
Concave Polygon Has at least one diagonal outside
the polygon
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Do Now
Classify Polygons as Convex or Concave
Classify the polygon below by its sides. Identify
it as convex or concave.
Starting with any side, count the number of sides
clockwise around the figure. Because the polygon
has 12 sides, it is a dodecagon.
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Triangle Angle-Sum Theorem
1. Draw and cut out a triangle. 2. Number the
angles and tear them off. 3. Place the angles
adjacent to each other. 4. Compare your results
with others. What do you observe about the sum
of the angles of a triangle?
Triangle Angle-Sum Theorem The sum of the
measures of the angles of a triangle measure 180º.
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STOP HERE
Activity Explore the sum of the interior angles
of Convex Polygons
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Polygon Angle-Sum Theorem
Use the Triangle Angle-Sum Theorem to find the
sum of the measures of the angles of a polygon.
1. Sketch convex polygons with 4, 5, 6, 7, and 8
sides. Construct a table to record your data in
order to look for a pattern or rule to find the
sum of the measures of the angles of an
n-gon. 2. Divide each polygon into triangles by
drawing all diagonals that are possible from one
vertex. 3. Multiply the number of triangles
by 180 to find the sum of the measures of the
angles of each polygon.
Polygon Number sides (n) of Triangles Sum of Interior angle measures (___ 180 ___)
4 2 2 180 360
n
(n - 2) 180
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Polygon Angle-Sum Theorem
Theorem 3-9 Polygon Angle-Sum Theorem
The sum of the measures of the angles of an n-gon
is (n - 2) 180.
Find the sum of the measures of the angles of a
decagon.
A decagon has 10 sides, so n 10.
Sum (n 2)(180) Polygon Angle-Sum
Theorem
(10 2)(180) Substitute 10 for n.
8 180 Simplify.
1440
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Think
Polygon Angle-Sum Theorem
The sum of the measures of the angles of a given
polygon is 720. How can you use the Polygon
Angle-Sum Theorem to find the number of sides in
the polygon?
Sum (n 2) 180 Write the
Equation
720 (n 2) 180 Sub. In known
values
720 180n 360 Simplify
1080 180n Addition
Prop of EQ
6 n
Hexagon (6 sides)
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Use the Polygon Angle-Sum Theorem
Find m ? X in quadrilateral XYZW.
The figure has 4 sides, so n 4.
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Polygon Exterior Angle-Sum Theorem

• Equilateral Polygon
• all sides are congruent

• Equiangular Polygon
• all angles are congruent

• Regular Polygon
• is both equilateral and equiangular

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Real-world Connection
Below is a regular hexagon game board packaged in
a rectangular box. Explain how you know that all
the angles labeled 1 have equal measures.
The hexagon is regular, so all its angles are
congruent.
An exterior angle is the supplement of a
polygons angle because they are adjacent angles
that form a straight angle.
Because supplements of congruent angles are
congruent, all the angles marked 1 have equal
measures.