The Polygon Angle-Sum Theorems Worksheet With Answers Page 4
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5
EXAMPLE
EXAMPLE
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4
EXAMPLE
Using the Polygon Angle-Sum Theorem
EXAMPLE
Point out that the angles of a
R
T
polygon can be called interior
Algebra
Find m&Y in pentagon TVYMR at the right.
= 5.
135
angles. Also point out that the
Use the Polygon Angle-Sum Theorem for
n
exterior angles of a regular
m&T + m&V + m&Y + m&M + m R =
- 2)180
(5
M
polygon are congruent.
90 + 90 + m&Y + 90 + 135 = 540
Substitute.
m&Y + 405 = 540
Simplify.
Y
V
PowerPoint
m&Y = 135
Subtract 405 from each side.
Additional Examples
Quick Check
4
4
Pentagon ABCDE has 5 congruent angles. Find the measure of each angle.
108
4
4
Find m&X in quadrilateral
XYZW.
You can draw exterior angles at any vertex of a polygon. The figures below show
Y
that the sum of the measures of the exterior angles, one at each vertex, is 360. This
X
can be proved as a theorem in a way suggested in Exercise 46.
59
75
80
115
86
98
150
99
100°
130
W
Z
71
76 41
85
80 +150 + 130 = 360
115 + 75 +99+ 71 =360
86 + 59 + 98+ 41+ 76 = 360
5
5
Explain how you know that all
the angles labeled &1 in Example
Key Concepts
Theorem 3-15
Polygon Exterior Angle-Sum Theorem
5 have equal measures.
Sample:
3
Because the hexagon is regular,
The sum of the measures of the exterior angles of a
all its angles are congruent. An
polygon, one at each vertex, is 360.
2
4
exterior angle is the supplement
For the pentagon,
1
of a polygon’s angles, and
5
m&1 + m&2 + m&3 + m&4 + m&5 = 360.
supplements of congruent angles
all have the same measure.
Resources
An
equilateral polygon
has all sides congruent. An
equiangular polygon
has all
• Daily Notetaking Guide 3-5
L3
angles congruent. A
regular polygon
is both equilateral and equiangular.
• Daily Notetaking Guide 3-5—
L1
Adapted Instruction
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5
Real-World
Connection
EXAMPLE
EXAMPLE
Packaging
The game board at the right has the shape of a
Closure
regular hexagon. It is packaged in a rectangular box
outlined beneath it. The box uses four right triangles made
If the sum of the interior angles
of foam in its four corners. Find m&1 in each foam triangle.
of a polygon equals the sum
Find the measure of an angle of the hexagon first.
of the exterior angles, what
is the name of the polygon?
2
2
• A regular hexagon has 6 sides and 6 congruent angles.
1
1
quadrilateral
If each exterior
The sum of the measures of the interior angles
For: Regular Polygon Activity
angle of a regular polygon
=
- 2)180, or 720.
(6
Use: Interactive Textbook, 3-5
measures 30, how many sides
720
,
• The measure of one interior angle is
or 120.
6
does the polygon have?
12
• The measure of its adjacent exterior angle, &1, is
1
1
180 - 120, or 60.
2
2
Quick Check
5
5
a. Find m&1 by using the Polygon Exterior Angle-Sum Theorem.
60
b. Find m&2. Is &2 an exterior angle? Explain.
30; no, it is not formed by
extending one side of the polygon.
160
Chapter 3
Parallel and Perpendicular Lines
160
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