The Polygon Angle-Sum Theorems Worksheet With Answers Page 3
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8Guided Instruction
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Polygon Angle Sums
Activity
Have students work in pairs
to draw polygons and display
Activity:
The Sum of Polygon Angle Measures
the sum of their angles using
geometry software. Have students
You can use triangles and the Triangle Angle-Sum Theorem to find the
manipulate the polygons to see
sum of the measures of the angles of a polygon. Record your data in a
that the sum remains constant.
table like the one begun below.
Number of
Number of
Sum of the Interior
Alternative Method
Polygon
Sides
Triangles Formed
Angle Measures
Students can use inductive
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• 180
reasoning to write the sum of
the measures of the angles of an
n-gon as 180n - 360. Have them
• Sketch polygons with 4, 5, 6, 7, and 8 sides.
draw a hexagon and segments
from an interior point to each
• Divide each polygon into triangles by drawing all diagonals that are
vertex. Ask: How many triangles
possible from one vertex.
are there?
What is the sum
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• Multiply the number of triangles by 180 to find the sum of the
of the angle measures of all the
measures of the angles of each polygon.
triangles?
180 ? 6, or 1080
What
See back of book.
is the sum of the angle measures
Vocabulary Tip
1. Look for patterns in the table. Describe any that you find.
of the triangles drawn from an
Inductive Reasoning
Write a rule for the sum of the measures of the
An n-gon is a polygon
2.
interior point to the vertices of a
with n sides, where n can
The sum of the measures of the angles of an
angles of an n-gon.
polygon with n sides?
180n
be 3, 4, 5, 6,
n-gon is (n
2) 180.
Point out that the sum of the
angle measures around the
interior point is 360. Discuss why
By dividing a polygon with n sides into n - 2 triangles, you can show that
360 must then be subtracted
the sum of the measures of the angles of any polygon is a multiple of 180.
from 180n.
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Key Concepts
Error Prevention
EXAMPLE
EXAMPLE
Theorem 3-14
Polygon Angle-Sum Theorem
Some students may think the
The sum of the measures of the angles of an n-gon is (n - 2)180.
answer should be 15 ? 180.
Review the activity to correct
them.
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Finding a Polygon Angle Sum
EXAMPLE
EXAMPLE
PowerPoint
Find the sum of the measures of the angles of a 15-gon.
Additional Examples
= 15.
For a 15-gon,
n
Sum =
- 2)180
(n
Polygon Angle-Sum Theorem
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Find the sum of the measures
=
- 2)180
(15
Substitute.
Visit:
of the angles of a decagon.
1440
= 13 180
Web Code: aue-0775
Simplify.
= 2340
The sum of the measures of the angles of a 15-gon is 2340.
Quick Check
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a. Find the sum of the measures of the angles of a 13-gon.
1980
Critical Thinking
b.
The sum of the measures of the angles of a given polygon is 720.
How can you use Sum = (n - 2)180 to find the number of sides in the polygon?
You can solve the equation (n
2)180
720.
You will sometimes use algebra with the Polygon Angle-Sum Theorem to find
measures of polygon angles.
159
Lesson 3-5
The Polygon Angle-Sum Theorems
Quick Check
BC
CD
AB
BC
BCDE; sides:
,
,
ABCDE; sides:
,
,
DE
,
EB
; ': l EBC, l C,
CD
,
DE
,
EA
;
1. ABE; sides:
AB
,
BE
,
EA
;
': l A, l ABC, l C, l D,
l D, l DEB
angles: l A, l ABE,
l AED
l BEA
159
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