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GUIDED PRACTICE
Vocabulary Check
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Concept Check
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3 APPLY
G
B
1. Name an interior angle and an
exterior angle of the polygon
shown at the right.
ASSIGNMENT GUIDE
C
A
BASIC
™A, ™B, ™BCD, ™D, or ™AED;
™AEF, ™BCG, or ™DCH
F
E
H
D
Day 1: pp. 665–666 Exs. 6–40
even
Day 2: pp. 665–668 Exs. 7–41 odd,
49–56, 58–61, 63–73 odd
2. How many exterior angles are there in an n-gon? Are they all considered
when using the Polygon Exterior Angles Theorem? Explain.
2n (2 at each vertex); no, only one at each vertex
Find the value of x.
3.
120ⴗ
95
4.
120
AVERAGE
5.
Day 1: pp. 665–666 Exs. 6–40
even
Day 2: pp. 665–668 Exs. 7–43 odd,
49–61, 63–73 odd
45
105ⴗ
115ⴗ
xⴗ
105ⴗ
ADVANCED
xⴗ
xⴗ
Day 1: pp. 665–666 Exs. 6–40
even
Day 2: pp. 665–668 Exs. 7–41 odd,
43–45, 49–62, 63–73 odd
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 823.
BLOCK SCHEDULE
SUMS OF ANGLE MEASURES Find the sum of the measures of the interior
angles of the convex polygon.
6. 10-gon 1440°
10. 20-gon
3240°
7. 12-gon 1800°
11. 30-gon 5040°
8. 15-gon
9. 18-gon 2880°
2340°
12. 40-gon 6840°
13. 100-gon 17,640°
ANGLE MEASURES In Exercises 14–19, find the value of x.
14. 127
xⴗ
15. 101
113ⴗ
146ⴗ
106ⴗ
147ⴗ
80ⴗ
16. 80 102ⴗ
98ⴗ
120ⴗ
143ⴗ
124ⴗ
170ⴗ
125ⴗ xⴗ
130ⴗ
17. 108
xⴗ
18.
158ⴗ
xⴗ
xⴗ
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 6–16, 20,
21
Example 2: Exs. 17–19,
22–28
Example 3: Exs. 29–38
Example 4: Exs. 39, 40,
49, 50
Example 5: Exs. 51–54
about 128.6
20. A convex quadrilateral has interior angles that measure 80°, 110°, and 80°.
What is the measure of the fourth interior angle? 90°
21. A convex pentagon has interior angles that measure 60°, 80°, 120°, and 140°.
What is the measure of the fifth interior angle? 140°
DETERMINING NUMBER OF SIDES In Exercises 22–25, you are given the
measure of each interior angle of a regular n-gon. Find the value of n.
22. 144° 10
23. 120° 6
24. 140° 9
25. 157.5° 16
11.1 Angle Measures in Polygons
EXERCISE LEVELS
Level A: Easier
6–13
Level B: More Difficult
14–42, 47–61
Level C: Most Difficult
43–46, 62
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 8, 14,
20, 22, 26, 30, 34, 38, 51, 56. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 11
Resource Book, p. 25)
•
Transparency (p. 80)
19. 135
xⴗ
pp. 665–668 Exs. 6–41, 43, 49–61,
63–73 odd
COMMON ERROR
EXERCISES 22–25 Students
may make some algebraic errors
when solving these problems. Point
out that after they set up the equation, they should multiply both sides
by n.
665
665
42. According to the Polygon
Interior Angles Thm., the
sum of the measures of
the interior √ of a convex
polygon is dependent only
on the number of sides.
Therefore, the sums of the
measures of the interior
√ of any two polygons
with the same number of
sides are equal. It does
not matter whether one or
both of the polygons are
regular or whether the
two are similar.
FOCUS ON
APPLICATIONS
L
AL I
FE
39. The yellow hexagon is regular
with interior angles measuring
120° each; the yellow pentagons
each have two interior angles
that measure 90° and three interior angles that measure 120°; the
triangles are equilateral with all
interior angles measuring 60°.
40. The yellow decagon is regular
with each interior angle measuring 144°; the yellow pentagons
each have two interior angles
measuring 90°, two interior
angles measuring 108° and one
interior angle measuring 144°;
the red quadrilaterals have one
angle measuring 144° and three
angles measuring 72° each.
43. Draw all the diagonals of ABCDE
that have A as one endpoint.
The diagonals, A
苶C苶 and A
苶D
苶,
divide ABCDE into 3 †s . By the
Angle Addition Post., måBAE =
måBAC + måCAD + måDAE.
Similarly, måBCD = måBCA +
måACD and måCDE = måCDA +
måADE. Then the sum of the
measures of the interior ås of
ABCDE is equal to the sum of the
measures of ås of †ABC, †ACD,
and †ADE. By the † Sum Thm.,
the sum of the measures of each
† is 180°, so the sum of the measures of the interior ås of ABCDE
is 3 ⴢ 180° = (5 – 2) ⴢ 180°.
RE
COMMON ERROR
EXERCISES 29–38 Students
may use the formulas for interior
angles to do these exercises. Point
out that if they are using exterior
angles, they should be using the
formulas on page 663 not the ones
on page 662.
41. ™3 and ™8 are a linear
pair, so m™3 = 140°; ™2
and ™7 are a linear pair,
so m™7 = 80°; m™1 =
80° by the Polygon Interior
Angles Thm.; ™1 and ™6
are a linear pair, so
m™6 = 100°; ™4 and ™9
are a linear pair, as are
™5 and ™10, so m™9 =
m™10 = 70°.
STAINED GLASS
is tinted glass that
has been cut into shapes
and arranged to form a
picture or design. The pieces
of glass are held in place by
strips of lead.
44. Let A be a regular n-gon
and x° the measure of
each interior ™. By the
Polygon Interior Angles
Thm., the sum of the
measures of the interior
√ of A is (n – 2) • 180°.
That is, n • x° =
(n – 2) • 180°, or x° =
(n º 2) • 180°
} }.
n
666
666
CONSTRUCTION Use a compass, protractor, and ruler to check the results
of Example 2 on page 662. 26–28. Check drawings.
26. Draw a large angle that measures 140°. Mark congruent lengths on the sides
of the angle.
27. From the end of one of the congruent lengths in Exercise 26, draw the second
side of another angle that measures 140°. Mark another congruent length
along this new side.
28. Continue to draw angles that measure 140° until a polygon is formed. Verify
that the polygon is regular and has 9 sides.
DETERMINING ANGLE MEASURES In Exercises 29–32, you are given the
number of sides of a regular polygon. Find the measure of each exterior
angle.
29. 12
30°
30. 11
about 32.7° 31. 21
about 17.1° 32. 15
24°
DETERMINING NUMBER OF SIDES In Exercises 33–36, you are given the
measure of each exterior angle of a regular n-gon. Find the value of n.
33. 60°
6
34. 20° 18
35. 72°
36. 10°
5
36
37. A convex hexagon has exterior angles that measure 48°, 52°, 55°, 62°, and
68°. What is the measure of the exterior angle of the sixth vertex? 75°
38. What is the measure of each exterior angle of a regular decagon? 36°
STAINED GLASS WINDOWS In Exercises 39 and 40, the purple and
green pieces of glass are in the shape of regular polygons. Find the
measure of each interior angle of the red and yellow pieces of glass.
39, 40. See margin.
39.
40.
41. FINDING MEASURES OF ANGLES
In the diagram at the right, m™2 = 100°,
m™8 = 40°, m™4 = m™5 = 110°.
Find the measures of the other labeled
angles and explain your reasoning.
See margin.
42.
8
9
7
2
3
4
5
1 6
10
Writing Explain why the sum of the measures of the interior angles of any
two n-gons with the same number of sides (two octagons, for example) is the
same. Do the n-gons need to be regular? Do they need to be similar?
See margin.
43.
PROOF Use ABCDE to write a paragraph
proof to prove Theorem 11.1 for pentagons.
See margin.
44.
PROOF Use a paragraph proof to prove
the Corollary to Theorem 11.1.
Chapter 11 Area of Polygons and Circles
A
E
B
C
D
45. Let A be a convex n-gon.
Each interior ™ and one
of the exterior √ at that
vertex form a linear pair,
so the sum of their
measures is 180°. Then
the sum of the measures
of the interior ™ and one
exterior ™ at each vertex
is n • 180°. By the Polygon
Interior Angles Thm., the
sum of the measures of
the interior √ of A is
(n – 2) • 180°. So the sum
of the measures of the
exterior √ of A, one at
each vertex, is n • 180° –
(n – 2) • 180° = n • 180° –
n • 180° + 360° = 360°.
45.
PROOF Use this plan to write a paragraph proof of Theorem 11.2.
Plan for Proof In a convex n-gon, the sum of the measures of an interior
(n – 2) ⴢ 180°
n
1
n = 3 }}. It is not possible for a
3
1
polygon to have 3 }} sides.
3
(n – 2) ⴢ 180°
54. No; if }} = 18°, then
n
2
n = 2 }}. It is not possible for a
9
2
polygon to have 2 }} sides.
9
53. No; if }} = 72°, then
angle and an adjacent exterior angle at any vertex is 180°. Multiply by n to
get the sum of all such sums at each vertex. Then subtract the sum of the
interior angles derived by using Theorem 11.1.
PROOF Use a paragraph proof to prove the Corollary to Theorem 11.2.
See margin.
TECHNOLOGY In Exercises 47 and 48, use geometry software to
construct a polygon. At each vertex, extend one of the sides of the
polygon to form an exterior angle.
46.
47. Measure each exterior angle and verify that the sum of the measures is 360°.
Check results.
48. Move any vertex to change the shape of your polygon. What happens to the
measures of the exterior angles? What happens to their sum? See margin.
49.
HOUSES Pentagon ABCDE is
an outline of the front of a house.
Find the measure of each angle.
50.
TENTS Heptagon PQRSTUV
is an outline of a camping tent.
Find the unknown angle measures.
46. Let A be a regular convex
n-gon. Since all the
m™A = m™E = 90°, m™B = m™C =
m™P = m™V = 70°, m™S = 140°
S
interior √ are £ and each
m™D = 120°
C
interior ™ and one of the
2xⴗ
R
T
150ⴗ
150ⴗ
exterior √ at that vertex
form a linear pair, the
q 160ⴗ
160ⴗ U
B
D
exterior √ are also £ by
the Congruent
xⴗ
xⴗ
Supplements Thm. Let x°
P
V
A
E
be the measure of each
exterior ™. Then n • x° = POSSIBLE POLYGONS Would it be possible for a regular polygon to have
360°
interior angles with the angle measure described? Explain. 51–54. See margin.
360° and x° = }}
55. ƒ(n ) is the measure of each
interior å of a regular n-gon;
as n gets larger and larger,
ƒ(n ) increases, becoming closer
and closer to 180°.
56. ƒ(n ) is the measure of each
exterior å of a regular n-gon;
as n gets larger and larger,
ƒ(n ) gets smaller and smaller.
n.
51. 150°
52. 90°
53. 72°
54. 18°
xy USING ALGEBRA In Exercises 55 and 56, you are given a function and its
graph. In each function, n is the number of sides of a polygon and ƒ(n) is
measured in degrees. How does the function relate to polygons? What
happens to the value of ƒ(n) as n gets larger and larger? 55, 56. See margin.
48. If the measure of an
180n º 360
interior ™ increases, the 55. ƒ(n) = ᎏnᎏ
measure of the corresp.
exterior ™ decreases. If
ƒ(n)
the measure of an interior
120
™ decreases, the measure
90
of the corresp. exterior ™
increases. The sum of the
60
measures of the exterior
30
√ is always 360°.
0
(n – 2) • 180°
51. Yes; if }} = 150°,
n
0
360
56. ƒ(n) = ᎏᎏ
n
ƒ(n)
120
90
60
30
0
3 4 5 6 7 8 n
0
For Lesson 11.1:
then n = 12. A regular 12LOGICAL REASONING You are
gon (dodecagon) has inte- 57.
rior √ with measure 150°.
shown part of a convex n-gon. The
52. Yes; a square is a regular
polygon with interior √
of 90°.
pattern of congruent angles continues
around the polygon. Use the Polygon
Exterior Angles Theorem to find the
value of n. 10
ADDITIONAL PRACTICE
AND RETEACHING
3 4 5 6 7 8 n
163ⴗ
125ⴗ
11.1 Angle Measures in Polygons
667
• Practice Levels A, B, and C
(Chapter 11 Resource Book,
p. 14)
• Reteaching with Practice
(Chapter 11 Resource Book,
p. 17)
•
See Lesson 11.1 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
667
Test
Preparation
4 ASSESS
QUANTITATIVE COMPARISON In Exercises 58–61, choose the statement
that is true about the given quantities.
A
¡
B
¡
C
¡
D
¡
DAILY HOMEWORK QUIZ
Transparency Available
1. Find the sum of the measures
of the interior angles in a
convex 16-gon. 2520°
2. Find the value of x.
156°
The two quantities are equal.
The relationship cannot be determined from the given information.
Column B
58.
The sum of the interior angle
measures of a decagon
The sum of the interior angle
measures of a 15-gon
B
59.
The sum of the exterior angle
measures of an octagon
8(45°)
C
60.
m™1
m™2
45°
124°
The quantity in column B is greater.
Column A
125°
x°
The quantity in column A is greater.
118ⴗ
3. The measure of each interior
angle of a regular polygon is
162°. How many sides does
the polygon have? 20
4. The measure of each exterior
angle of a regular polygon is
6°. How many sides does the
polygon have? 60
1
61.
★ Challenge
135ⴗ
91ⴗ
156ⴗ
2
111ⴗ
146ⴗ
Number of sides of a polygon
with an exterior angle
measuring 72°
A
70ⴗ
72ⴗ
D
Number of sides of a
polygon with an exterior
angle measuring 144°
Æ
Æ
62. Polygon STUVWXYZ is a regular octagon. Suppose sides ST and UV are
extended to meet at a point R. Find the measure of ™TRU. 90°
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 11.1 are available in
blackline format in the Chapter 11
Resource Book, p. 22 and at
www.mcdougallittell.com.
MIXED REVIEW
FINDING AREA Find the area of the triangle described. (Review 1.7 for 11.2)
63. base: 11 inches; height: 5 inches
64. base: 43 meters; height: 11 meters
236.5 m2
27.5 in.2
65. vertices: A(2, 0), B(7, 0), C(5, 15)
66. vertices: D(º3, 3), E(3, 3), F(º7, 11)
37.5 sq. units
24 sq. units
VERIFYING RIGHT TRIANGLES Tell whether the triangle is a right triangle.
(Review 9.3)
ADDITIONAL TEST
PREPARATION
1. WRITING Explain how to find
the sum of the measures of the
interior angles of a polygon
using triangles. Use diagonals to
67. no
divide the polygon into triangular
regions. The sum of the angles in
each triangle is 180°. Multiply the
number of triangles by 180° to get
the sum of the measures of the
interior angles in the polygon.
68. yes
69. no
75
21
16
13
7
5
72
2兹17
9
Æ
Æ
FINDING MEASUREMENTS GD and FH are diameters of
circle C. Find the indicated arc measure. (Review 10.2)
៣
៣
72. mEH
70. mDH 80°
145°
៣ 65°
២ 245°
73. mEHG
71. mED
G
F
35ⴗ
C
80ⴗ
E
D
668
668
Chapter 11 Area of Polygons and Circles
H