Download Decomposing Polygons

Decomposing Polygons: Interior
and Exterior Angles
Do Now: Complete the following list:
Triangle: 3
_______________: 8
_______________: 7
Undecagon: ___
_______________: 4
Hexagon: ___
______________: 10
Nonagon:___
______________: 12
Pentagon: ___
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In geometry, it is necessary to know the sum of the interior angles of various
polygons to determine other information. An interior angle of a polygon is an
angle which faces the inside of a polygon and is formed by consecutive sides of the
polygon. Is there a quick method for calculating the sum of the measures of different
polygons? Let’s find out!
Ms. Lambert asked her class to determine the sum of the interior angle measures
of a quadrilateral. She that two of her students, Carson and Juno, were already
engaged in a heated disagreement.
Lesson 8.4
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Decomposing Polygons
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Carson drew a quadrilateral and added one diagonal as shown. He concluded
that the sum of the measures of the interior angles of a quadrilateral must be
equal to 360º.
1. Describe Carson’s reasoning.
Juno drew a quadrilateral and added two diagonals as shown. She concluded
that the sum of the measures of the interior angles of a quadrilateral must be equal
to 720º.
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2. Describe Juno’s reasoning.
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As always, you must start with what you know to be true. The Triangle Sum Theorem
states that the sum of the three interior angles of any triangle is equal to 180o. You can
use this information to calculate the sum of the interior angles of other polygons.
Calculate the sum of the interior angle measures of a quadrilateral by completing each
step.
Step 1: Draw a quadrilateral.
Step 2: Draw all possible diagonals using only one vertex of the quadrilateral.
Remember, a diagonal is a line segment connecting non-adjacent vertices.
Step 3: How many triangles are formed when the diagonal(s) divide the quadrilateral?
Step 4: If the sum of the interior angle measures of each triangle is 180, what is the sum
of all the interior angle measures of the triangles formed by the diagonal(s)?
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Calculate the sum of the interior angle measures of a pentagon by completing each step.
Step 1: Draw a pentagon.
Step 2: Draw all possible diagonals using only one vertex of the pentagon.
Step 3: How many triangles are formed when the diagonal(s) divide the pentagon?
Step 4: If the sum of the interior angle measures of each triangle is 180, what is the sum of
all the interior angle measures of the triangles formed by the diagonal(s)?
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Calculate the sum of the interior angle measures of a hexagon by completing each step.
Step 1: Draw a hexagon.
Step 2: Draw all possible diagonals using one vertex of the hexagon.
Step 3: How many triangles are formed when the diagonal(s) divide the hexagon?
Step 4: If the sum of the interior angle measures of each triangle is 180, what is the sum
of all the interior angle measures of the triangles formed by the diagonal(s)?
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4. Complete the table shown.
Number of sides of the polygon
3
4
5
6
Number of diagonals drawn
8
Number of triangles formed
Sum of the measures of the interior angles
5. What pattern do you notice between the number of possible diagonals drawn
from one vertex of the polygon, and the number of triangles formed by those
diagonals?
6. Compare the number of sides of the polygon to the number of possible
diagonals drawn from one vertex. What do you notice?
7. Compare the number of sides of the polygon to the number of triangles formed
by drawing all possible diagonals from one vertex. What do you notice?
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8. What pattern do you notice about the sum of the interior angle measures of a
polygon as the number of sides of each polygon increases by 1?
9. Predict the number of possible diagonals drawn from one vertex and the
number of triangles formed for a seven-sided polygon using the table you
completed.
10. Predict the sum of all the interior angle measures of a seven-sided polygon
using the table your completed.
11. Continue the pattern to complete the table.
Number of sides of the polygon
7
8
9
16
Number of diagonals drawn
Number of triangles formed
Sum of the measures of the interior angles
12. When you calculated the number of triangles formed in the 16-sided polygon,
did you need to know how many triangles were formed in a 15-sided polygon
first? Explain your reasoning.
13. If a polygon has 100 sides, how many triangles are formed by drawing all
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14. What is the sum of all the interior angle measures of a 100-sided polygon?
Explain your reasoning.
15. If a polygon has n sides, how many triangles are formed by drawing all
diagonals from one vertex? Explain.
© 2010 Carnegie Learning, Inc.
13. If a polygon has 100 sides, how many triangles are formed by drawing all
possible diagonals from one vertex? Explain.
16. What is the sum of all the interior angle measures of an n-sided polygon?
Explain your reasoning.
17. Use the formula to calculate the sum of all the interior angle measures of a
polygon with 32 sides.
Quadrilaterals
18. If the sum of all the interior angle measures of a polygon is 9540º,
how many sides does the polygon have? Explain your reasoning.
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of a Regular Polygon
1. Use the formula developed in Problem 2, Question 16 to calculate the sum of
the all the interior angle measures of a decagon.
2. Calculate each interior angle measure of a decagon if each interior angle is
congruent. How did you calculate your answer?
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3. Complete the table.
Number of sides of regular polygon
3
4
5
6
7
8
Sum of measures of interior angles
Measure of each interior angle
4. If a regular polygon has n sides, write a formula to calculate the measure of
each interior angle.
5. Use the formula to calculate each interior angle measure of a regular
100-sided polygon.
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Decomposing Polygons
Decomposing
Polygons
Sum of the Interior
Angle Measures of a Polygon
Home Work
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Sum of the Interior Angle Measures of a Polygon
Determine the measure of an interior angle of the given regular polygon.
Determine
the measure of an interior
angle of the given regular polygon.
1. regular nonagon
2. regular decagon
1. regular nonagon
2. regular decagon
3. regular 15-gon
4. regular 47-gon
3. regular 15-gon
4. regular 47-gon
Determine the measure of the missing angle in each figure.
5.
121°
135°
Determine the measure90°of the missing angle in each figure.
108°
© 2010 Carnegie Learning,
Inc.
© 2010 Carnegie Learning, Inc.
5.
121°
135°
166°
?
90°
108°
166°
6.
135°
?
?
146°
142°
6.
113°
128°
135°
161°
99°
?
146°
142°
11
128°
7. Use the figure to answer each question.
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Q
7x – 12
P
4x
R
2x
4x + 2
S
5x
T
a. What is the sum of the measures of the interior angles of the polygon?
b. What is the value of x?
d. What is the measure of angle !RQP?
© 2010 Carnegie Learning, Inc.
c. What is the measure of !PTS?
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