Download 11.1 Angle Measures in Polygons

Pre-AP Bellwork 10-19
3) Solve for x..
30
(8 + 6x)
(4x + 2)°
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Pre-AP Bellwork 10-24
5) Find the values of the variables and then the
measures of the angles.
z°
x°
w°
y°
30°
(2y – 6)°
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3-4 Polygon Angle-Sum
Theorem
Geometry
3
Q
VERTEX
R
SIDE
Definitions:
P
S
VERTEX
T

Polygon—a plane figure that meets
the following conditions:



It is formed by 3 or more segments
called sides, such that no two sides with
a common endpoint are collinear.
Each side intersects exactly two other
sides, one at each endpoint.
Vertex – each endpoint of a side.
Plural is vertices. You can name a
polygon by listing its vertices
consecutively. For instance, PQRST
and QPTSR are two correct names for
the polygon above.
Example 1: Identifying Polygons




State whether the
figure is a polygon.
If it is not, explain
why.
Not D – has a side
that isn’t a
segment – it’s an
arc.
Not E– because two
of the sides
intersect only one
other side.
Not F because
some of its sides
intersect more than
two sides/
A
C
B
F
E
D
Figures A, B, and C are
polygons.
Polygons are named by the number of sides they have –
MEMORIZE
Number of sides
3
Type of Polygon
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
9
10
12
n
Octagon
Nonagon
Decagon
Dodecagon
n-gon
Convex or Concave???
A convex polygon has
no diagonal with points
outside the polygon.
A concave polygon
has at least one
diagonal with points
outside the polygon
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Measures of Interior and Exterior
Angles

You have already learned the name
of a polygon depends on the
number of sides in the polygon:
triangle, quadrilateral, pentagon,
hexagon, and so forth. The sum of
the measures of the interior angles
of a polygon also depends on the
number of sides.
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Measures of Interior and Exterior
Angles

For instance . . . Complete this table
Polygon
Triangle
# of
sides
3
Quadrilateral
# of
triangles
1
Sum of measures of
interior ’s
1●180=180
2●180=360
Pentagon
Hexagon
Nonagon (9)
n-gon
n
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Pre-AP Bellwork 10 - 24
6) Find the sum of the interior angles of
a dodecagon.
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Measures of Interior and Exterior
Angles


What is the pattern?
(n – 2) ● 180.
This relationship can be used to find
the measure of each interior angle
in a regular n-gon because the
angles are all congruent.
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Ex. 1: Finding measures of Interior
Angles of Polygons

Find the value of x
in the diagram
shown:
142
88
136
105
136
x
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SOLUTION:


S(hexagon)=
(6 – 2) ● 180 = 4
● 180 = 720.
Add the measure
of each of the
interior angles of
the hexagon.
142
88
136
105
136
x
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SOLUTION:
136 + 136 + 88 +
142 + 105 +x =
720.
607 + x = 720
X = 113
•The measure of the sixth interior angle of
the hexagon is 113.
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Polygon Interior Angles Theorem

The sum of the
measures of the
interior angles of a
convex n-gon is
(n – 2) ● 180
COROLLARY:
The measure of
each interior
angle of a
regular n-gon is:

1
n
or
● (n-2) ● 180
( n  2)(180)
n
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EX.2 Find the measure of an
interior angle of a decagon….
n=10
(n  2)(180)

n
(10  2)(180)

10
8(180)

10
 144
 144
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Ex. 2: Finding the Number of Sides of
a Polygon


The measure of each interior angle
is 140. How many sides does the
polygon have?
USE THE COROLLARY
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Solution:
( n  2)(180)
n
= 140
(n – 2) ●180= 140n
Corollary to Thm. 11.1
Multiply each side by n.
180n – 360 = 140n
Distributive Property
40n = 360
Addition/subtraction
props.
n = 90
Divide each side by 40.
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Copy the item below.
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EXTERIOR ANGLE THEOREMS
3-10
3-10
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Ex. 3: Finding the Measure of an
Exterior Angle
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Ex. 3: Finding the Measure of an
Exterior Angle
3-10
Simplify.
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Ex. 3: Finding the Measure of an
Exterior Angle
3-10
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Using Angle Measures in Real Life
Ex. 4: Finding Angle measures of a polygon
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Using Angle Measures in Real Life
Ex. 5: Using Angle Measures of a Regular
Polygon
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Using Angle Measures in Real Life
Ex. 5: Using Angle Measures of a Regular
Polygon
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Using Angle Measures in Real Life
Ex. 5: Using Angle Measures of a Regular
Polygon
Sports Equipment: If you were
designing the home plate marker
for some new type of ball game,
would it be possible to make a
home plate marker that is a regular
polygon with each interior angle
having a measure of:
a. 135°?
b. 145°?
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Using Angle Measures in Real Life
Ex. : Finding Angle measures of a polygon
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