Name: _____________________________________________________ Date: ____________ Period: ___
Shapes & Designs Investigation 2.1: Angle Sums of Regular Polygons
 Use p. 41 of your book (or the glossary) to define the following words.
Regular Polygon: _______________________________________________________________________
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Irregular Polygon: ______________________________________________________________________
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 Use an angle ruler to measure the angles in the regular polygons from the
Shapes Set on page 42 of your book.
1) Enter the results in the table below.
Polygon
Number of
Sides
Sum of All of the
Measure of 1 Angle
Angles Measures in
in the Figure
the Figure
Triangle
Square
Pentagon
Hexagon
Heptagon
Octagon
2) Find a pattern in the table above and use the pattern to fill in the number of sides, angle
measure, and angle sum for the Nonagon and the Decagon.
Polygon
Nonagon
Decagon
Number of
Sides
Measure of 1 Angle
in the Figure
Sum of All of the
Angle Measures in
the Figure
3) Look at the “Number of Sides” column and the “Sum of All of the Angles” column in the
table that you filled in. What rule can you create to calculate the sum of the angles in a
figure with ANY number of given sides?
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Shapes & Designs Investigation 2.2: Angle Sums of Any Polygon
We can use the following method to determine the angle sum of any polygon.
 Divide the polygon up into triangles by drawing diagonals from ONE vertex.
Example: You are able to draw 1 diagonal from one vertex of a quadrilateral.
Notice that by drawing the one diagonal,
2 triangles were formed.
1 triangle = 180⁰ so,
2 triangles = _________⁰
Therefore, there are __________⁰ in a quadrilateral.
 Divide the polygons below up into triangles by drawing diagonals from ONE
vertex. Complete the table below.
Polygon
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Decagon
Number of
Sides
Number of
Triangles Formed
when Diagonals
are Drawn from 1
Vertex
4
2
Sum of All of the
Angle Measures in
the Figure
 Based on what you know about the total angle measures in polygons, find the
missing angle measure in the polygons below.
For each problem, set-up an EQUATION and show all of your steps to solve for the missing
angle.
1)
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2)
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3)
4)
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For #s 5-7, don’t forget to set-up an EQUATION and show all of your steps to solve for the
missing angle. You can draw the picture if it helps you.
5) In a pentagon, each of two angles has a measure of 68°. Each of two other
angles measures 142°. What is the measure of the remaining angle?
6) In a quadrilateral, one angle measures 175°. A second angle measures 29°,
while a third angle measures 54°. What is the measure of the remaining
angle?
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7) In a hexagon, each of three angles has a measure of 83°. Two of the other
angles each have a measure of 150°. What is the measure of the remaining
angle?