Download Lesson 6A: Interior Angles of Polygons

Lesson 6A: Interior Angles of Polygons
1. Explore!

How can we use the fact that the sum of the angles
of a triangle equals 180° to find the sum of the
angles of the nonagon below?

Is there a formula we could use?
Think! What do the angle measurements of the polygon below add up to? ______________
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2. What’s the formula???



Draw enough diagonals to divide each polygon into triangles.
The vertices of all the triangles must be vertices of the polygon as
well.
Record the number of sides that the polygon has and the sum of its
interior angle measurements. (Hint: what is the sum of angle
measurements for each triangle?)
a) Quadrilaterals
n=_____________
b) Pentagons
Angle Sum:______________
c) Hexagons
n=_____________
n=_____________
Angle Sum:______________
d) Heptagons
Angle Sum:______________
n=_____________
Angle Sum:______________
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e) Octagons
N = _________________
Angle Sum = ____________
f) Nonagons
N = _________________
Angle Sum = ____________
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
Use your answers to complete the table
below.

In order to fill in all of the blanks, you will
have to look for patterns.
Number of sides the
Polygon has
(n)
Number of triangles
it was divided into
Angle Sum
(S)
*******
******
3
4
5
6
7
8
9
10
11
12
13
********
20
45
n
What’s the formula?
S=_____________________________
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3. Use it!

Find the missing angle measurements in the figures below. Start by deciding what the sum of the
angles should be for the given shape.
?
135°
?
59°
111°
42°
47°
104°
?
80°
?
?
38°
142°
262°
142°
70°
58°
?
43°
?
266°
32°
?
274°
69°
34°
50°
34°
64°
63°
4. Fill in the blank.
a) The sum of the interior angles of a polygon with 18 sides is _____________.
b) The sum of the interior angles of a polygon with 23 sides is _____________.
c) The angles of a certain polygon are all congruent. If it has 16 sides, what is the measure of each
angle? ______________
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5. Regular Polygons
Complete the definition:
A regular polygon is…
Vocabulary Word
The polygons below are all regular. Find the missing angle measurements.
6. More blanks…
a) A regular polygon has 14 sides. What are the measures of its interior angles? __________
b) A regular polygon has 20 sides. What are the measures of its interior angles? __________
c) What are the measures of the interior angles of a regular triangle? _________
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7. Algebra Time.

a)
Determine the value of x in each of the figures below. Use your answers to find the measures of all the
angles.
b)
(2x+3)°
(3x)°
(2x-6)°
(6x+17)°
(4x)°
(7x-20)°
Equation and Solution:
Equation and Solution:
Angle Measures:________________________
Angle Measures:________________________
c)
d)
(3x-22)°
21°
40°
(5x-2)°
324°
(2x+4)°
30°
(3x-22)°
(4x+5)°
Equation and Solution:
Equation and Solution:
Angle Measures:________________________
Angle Measures:________________________
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8. Ratios are back…

Find the angle measurements of the polygons described
below.
1:2:4:5:7
a) A quadrilateral with angle measurements whose
ratios are 1:2:6:6.
b) A quadrilateral with two right angles and the other
two angle measurements have ratio 1:3.
Sketch:
Sketch:
Equation and Solution:
Equation and Solution:
Angle Measures:________________________
Angle Measures:________________________
c) A pentagon with angle measurements whose
ratios are 3:3:4:4:4
d) A hexagon whose angle measurements have ratios
1:1:1:2:2:5.
Sketch:
Sketch:
Equation and Solution:
Equation and Solution:
Angle Measures:________________________
Angle Measures:________________________
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