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NAME ________________________________________ DATE ______________ PERIOD _____
6-3
Lesson Reading Guide
Polygons and Angles
Get Ready for the Lesson
Read the introduction at the top of page 316 in your textbook.
Write your answers below.
1. Predict the number of triangles and the sum of the angle measures in a
polygon with 8 sides.
2. Write an algebraic expression that could represent the number of
triangles in an n-sided polygon. Then write an expression to represent the
sum of the angle measures in an n-sided polygon.
Read the Lesson
4. Why do you think that you need to subtract 2 from the number of sides?
Lesson 6–3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. How many triangles would be in a 12-sided polygon?
5. What do you call the angles that lie inside a polygon?
Remember What You Learned
6. The outside walls of a sports stadium create a giant regular 60-sided
figure. Write an equation to find the number of triangles inside the
figure. Then write and solve an equation to find the sum of the interior
angles of the figure.
Chapter 6
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Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
6-3
Study Guide and Intervention
Polygons and Angles
An interior angle is an angle with sides that are adjacent sides of the polygon. A regular polygon is a
polygon whose sides and angles are congruent.
Example 1
Find the sum of the measures of the interior angles of a
tricontagon, which is a 30-sided polygon.
S (n 2)180º
Write an equation.
S (30 2)180º
Replace n with 30. Subtract.
S (28)180º
Multiply.
S 5,040º
The sum of the measures of the interior angles of a tricontagon is 5,040º.
The defense department of the United States has its headquarters
in a building called the Pentagon because it is shaped like a
regular pentagon. What is the measure of an interior angle of a
regular pentagon?
S (n 2)180º
S (5 2)180º
S (3)180º
S 540º
540º 5 108º
Write an equation.
Replace n with 5. Subtract.
Multiply.
Divide by the number of interior angles to find the measure
of one angle.
The measure of one interior angle of a regular pentagon is 108º.
Exercises
For Exercises 1–6, find the sum of the measures of the interior angles
of the given polygon.
1. nonagon (9-sided)
2. 14-gon
3. 16-gon
4. hendecagon (11-sided)
5. 25-gon
6. 42-gon
For Exercises 7–12, find the measure of one interior angle of the
given regular polygon. Round to the nearest hundredth if necessary.
7. hexagon
9. 22-gon
8. 15-gon
10. icosagon (20-sided)
11. 38-gon
12. pentacontagon (50-sided)
Chapter 6
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Course 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2
NAME ________________________________________ DATE ______________ PERIOD _____
6-3
Skills Practice
Polygons and Angles
1. 13-gon
2. 17-gon
3. 18-gon
4. 24-gon
5. 32-gon
6. 35-gon
7. 21-gon
8. 29-gon
9. 54-gon
10. 64-gon
11. 81-gon
12. 150-gon
Find the measure of one interior angle of the given regular polygon.
Round to the nearest hundredth if necessary.
13. heptagon (7-sided)
14. 26-gon
15. decagon (10-sided)
16. 23-gon
17. 37-gon
18. 51-gon
19. 48-gon
20. 85-gon
21. 72-gon
22. 49-gon
23. 66-gon
24. 500-gon
Chapter 6
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Lesson 6-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the sum of the measures of the interior angles of each polygon.
Course 3
NAME ________________________________________ DATE ______________ PERIOD _____
6-3
Practice
Polygons and Angles
Find the sum of the measures of the interior angles of each polygon.
1. 13-gon
2. 16-gon
3. 17-gon
4. 18-gon
5. 20-gon
6. 25-gon
Find the measure of one interior angle in each regular polygon.
Round to the nearest tenth if necessary.
7. pentagon
8. hexagon
9. 24-gon
ALGEBRA For Exercises 10 and 11, determine the angle measures in
each polygon.
10.
x⬚
5x ⬚
x⬚
x⬚
11.
5x⬚
1.5x ⬚
1.5x ⬚
x⬚
x⬚
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
12. FLOORING A floor is tiled with a pattern consisting of
regular octagons and squares as shown. Find the measure
of each angle at the circled vertex. Then find the sum of
the angles.
13. ART Jose is laying out a pattern for a stained glass
window. So far he has placed the 13 regular polygons shown.
Find the measure of each angle at the circled vertex. Then
find the sum of the angles.
14. REASONING Vanessa’s mother made a quilt using a
pattern of repeating regular hexagons as shown. Will
Vanessa be able to make a similar quilt with a pattern
of repeating regular pentagons? Explain your reasoning.
Chapter 6
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NAME ________________________________________ DATE ______________ PERIOD _____
6-3
Word Problem Practice
Polygons and Angles
For Exercises 1–6, use the formula S (n 2)180 to solve.
1. FLOORING Martha’s kitchen floor is
made from a tessellation of rows of
regular octagons. The space between
them is filled with square tiles as
shown below. Find the measure of one
interior angle in both the octagon and
the square tiles.
2. CIRCLES As the number of sides of a
regular polygon increase, the polygon
gets closer and closer to a true circle.
The interior angles of any regular
polygon can never actually reach 180º.
How many sides would a polygon have
whose interior angles are exactly 179º?
3. GEOMETRY A trapezoid has angles that
measure 3x, 3x, x, and x. What is the
measure of x?
4. GEOMETRY An irregular heptagon has
angles that measure x, x, 2x, 2x, 3x,
3x, and 4x. What is the measure of x?
3x ⬚
2x
x⬚
x⬚
Lesson 6–3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3x ⬚
3x
x
4x
5. TILES A bathroom tile consists of
regular hexagons surrounded by
regular triangles as shown below. Find
the measure of one interior angle in
both the hexagon and the triangle tiles.
Chapter 6
6. CHALLENGE How many sides does a
regular polygon have if the measure of
an interior angle is 171º?
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NAME ________________________________________ DATE ______________ PERIOD _____
6-3
Enrichment
M.C. Escher
Maurits Cornelis Escher (1898-1972) was a Dutch graphic and mathematical
artist. Some of his most famous pieces used tessellations, or repeated tiling
of one or more shapes. His designs range from artfully simple to extremely
intricate.
A regular polygon will tessellate a plane if the measure of one of its
interior angles is a factor of 360°. Other combinations of polygons
tessellate if the sum of the measures of the adjoining angles equals
360. The tessellation at the right is made of regular octagons and
squares. At any vertex the sum of the measures of the angles is
90 135 135 or 360.
1. Make a list of all regular polygons that will tessellate.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. Explain why you know there are no other regular polygons that
will tessellate.
For Steps A–E, you will create your own tessellation on a separate
piece of paper.
Step A Start with a polygon that will tessellate. Trace it,
and cut it out. Grid paper or isometric dot paper
may help you accurately draw your shape.
Step A
Step B Cut a portion of the figure on one side, slide it to
the opposite side, and tape it on. (This was done
three times in the example at right.)
Step B
Step C Use the modified shape as a tracing template.
Trace the template on another sheet of paper.
Step C
Step D Slide, reflect, and/or rotate the shape so that it
fits with your first tracing. Trace the template
where it fits with the previous tracing. Repeat
the process to cover the page.
Step E Color each polygon in the tessellation. Escher
often decorated the shapes so that they
resembled objects or animals.
Chapter 6
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Course 3