Download Polygon Angle Sum Conjectures

8.1 Find Angle Measures in Polygons
4/4/2013
8.1 Introduction
What is the sum of the interior angles in the
polygon below?
U
M
S
8.1 Introduction
What’s the difference between convex and
concave polygons?
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8.1 Find Angle Measures in Polygons
4/4/2013
Investigation 1
Using the two previous concepts, we will discover a
method for finding the sum of the angles in any
convex n-gon, where n is the number of sides (or
angles) of a given polygon.
Step 1: We start by drawing a series of convex
n-gons, starting with n = 3 and ending with n = 6.
Investigation 1
180
180
180
180
180
180
180
180
180
180
Step 2: In each polygon, if we draw all of the
diagonals from one vertex, notice how
these diagonals divide the polygons into
triangles. How could this help find the sum
of the angles in each n-gon?
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8.1 Find Angle Measures in Polygons
4/4/2013
Investigation 1
180
180
180
180
180
180
180
180
180
180
Step 3: Complete the table.
Number of
sides
3
4
5
6
Number of
triangles
Angle Sum
Polygon Interior Angles Theorem
Step 4: Find a formula.
The sum of the measures of the interior
angles of a convex n-gon is (n – 2)·180°.
m1 + m2 + … + mn = (n – 2)·180°
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8.1 Find Angle Measures in Polygons
4/4/2013
Example 3
What is the sum of the measures of the
interior angles of a convex octagon?
Example 4
What is the measure of each angle of an
equiangular octagon?
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8.1 Find Angle Measures in Polygons
4/4/2013
Example 5
Find the values of e and f.
Equiangular Polygon Theorem
The measure of each angle of an
equiangular n-gon can be found by using
either of the following expressions:
(n  2)  180
360
or 180 
n
n
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8.1 Find Angle Measures in Polygons
4/4/2013
Example 6
In a regular polygon, the measure of each
angle is 150˚. How many sides does the
polygon have?
Example 7: SAT
If the degree measures of the angles of a
quadrilateral are 4x, 7x, 9x, and 10x, what
is the sum of the measures of the smallest
angle and the largest angle?
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8.1 Find Angle Measures in Polygons
4/4/2013
Investigation 2
When you extend one
side of a triangle, you
form an exterior
angle. If you extend
each side of a
polygon to form one
exterior angle at each
vertex, you create a
set of exterior angles
for the polygon.
Polygon Exterior Angles Theorem
The sum of the measures of one set of
exterior angles of a polygon is 360°.
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8.1 Find Angle Measures in Polygons
4/4/2013
Example 8
What is the value of x?
Example 9
What is the number of sides of a polygon in
which the sum of the degree measures of
the interior angles is 4 times the sum of the
degree measures of the exterior angles?
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8.1 Find Angle Measures in Polygons
4/4/2013
Example 10
What is the measure of each exterior angle
in an equiangular octagon?
What is the measure of each exterior angle
in an equiangular n-gon? How does this
relate to the Equiangular Polygon
Theorem?
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