Transcript Slide 1

Geometry
Properties and
Attributes of Polygons
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Warm up
Solve by factoring:
1) x2 + 3x – 10 = 0
2) x2 - x – 12 = 0
3) x2 - 12x = - 35
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Properties and Attributes of Polygons
Today you will learn about the parts of polygon and the ways
to classify polygons.
Each segment that forms a polygon is a side of the polygon.
The common endpoint of two sides is a vertex of the
polygon. A segment that connects any two nonconsecutive
vertices is a diagonal.
A
B
side
E
D
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C
vertex
diagonal
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You can name a polygon by the number of its sides. The
table shows the names of some common polygons.
Polygon ABCDE in the previous slide is a pentagon.
Number of Slides
Name of Polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
n - gon
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Identifying Polygon
Tell whether each figure is a polygon. If it is a
polygon, name it by the number of its sides:
Polygon
Pentagon
Not a Polygon
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Polygon
Octagon
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Now you try!
Tell whether each figure is a polygon. If it is a polygon,
name it by the number of its sides:
1a
1b
1c
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All the sides are congruent in an equilateral polygon. All the
angles are congruent in an equiangular polygon. A regular
polygon is one that is both equilateral and equiangular. If a
polygon is not regular, it is called irregular.
A polygon is concave if any part of a diagonal contains points
in the exterior of the polygon. If no diagonal contains points in
the exterior, then the polygon is convex.
concave
quadrilateral
convex
quadrilateral
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Classifying Polygons
Tell whether each polygon is regular or irregular.
Tell whether it is concave or convex:
A
irregular
convex
Next page ->
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Tell whether each polygon is regular or irregular.
Tell whether it is concave or convex:
C
B
irregular
concave
regular
convex
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Now you try!
Tell whether each polygon is regular or irregular.
Tell whether it is concave or convex:
2a
2b
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To find the sum of the interior angles measure of a convex
polygon, draw all possible diagonals from one vertex of the
polygon. This creates a set of triangles.
The sum of the angle measures of all the triangles
equals the sum measures of the polygon.
Quadrilateral
Triangle
Pentagon
Hexagon
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Polygon
Number of
Slides
Number of
Triangles
Sum of Interior
Angle Measures
Triangle
3
1
(1) 180° = 180°
Quadrilateral
4
2
(1) 180° = 360°
Pentagon
5
3
(1) 180° = 540°
Hexagon
6
4
(1) 180° = 720°
n - gon
n
n-2
(n - 2) 180°
In each convex polygon, the number of triangles formed is
two less than the number of sides n. So the sum of the
angle measures of all these triangles is (n - 2) 180°.
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Polygon Angle Sum Theorem
The sum of the interior angle measures of a convex
polygon with sides n is
(n - 2) 180°.
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Finding Interior Angle Measures
and Sums in Polygons
A) Find the sum of the interior angle measures of a
convex octagon.
(n - 2) 180°
Polygon ∕ Sum thm.
= (8 - 2) 180°
An octagon has 8 sides. So,
substitute 8 for n.
= 1080°
Simplify.
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Finding Interior Angle Measures
and Sums in Polygons
B) Find the measure of each interior angle of a regular nonagon.
Step1: Find the sum of the interior angle measures.
(n - 2) 180°
Polygon ∕ Sum thm.
= (9 - 2) 180°
Substitute 9 for n.
= 1260°
Simplify.
Step2: Find the measure of one interior angle.
1260° = 140°
9
The int. ∕s are congruent, so divide by 9.
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Finding Interior Angle Measures
and Sums in Polygons
C) Find the measure of each interior angle
of a quadrilateral PQRS.
Q
3c°
P c°
(4 - 2) 180° = 360°
Polygon ∕ Sum thm.
m∕P + m∕Q + m∕R + m∕S
= 360°
Polygon ∕ Sum thm.
c + 3c + c + 3c = 360°
Substitute.
8c = 360°
R
c°
3c° S
=> c = 45°
m∕P =m∕R = 45°
m∕Q = m∕S = 360°
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Now you try!
3a) Find the sum of the interior angle measures of a
convex 15 - gon.
3b) Find the measure of each interior angle of a regular
decagon.
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In the polygons below, an exterior angle has been
measured at each vertex. Notice that in each case, the
sum of the exterior angle measure is 360°.
81°
147°
132°
41°
147° + 81° + 132° = 360°
111°
43°
55°
110°
43° + 111° + 41° + 55°
+ 110° = 360°
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Polygon Exterior Angle Sum
Theorem
The sum of the exterior angle measures , one angle at
each vertex, of a convex polygon with sides n is
360°.
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Finding Exterior Angle Measures in Polygons
A) Find the measure of each exterior angle of a regular hexagon.
A hexagon has 6 sides and 6 vertices.
Sum of the exterior angle = 360°
Polygon ext ∕ Sum thm.
Measure of one exterior angle
= 360° = = 60°
6
A regular hexagon has 6 ext ∕s.
So, divide the sum by 6.
The measure of each exterior angle of a regular
hexagon = 60°
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Finding Exterior Angle Measures in Polygons
B) Find the value of a in polygon RSTUV.
7a°
R
2a°
S
2a°
T
6a°
3a°
U
V
7a° + 2a° + 3a° + 6a° + 2a° = 360°
20a°= 360°
a = 60°
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Polygon ext ∕ Sum thm.
Combine like terms.
So, divide the sum by 20.
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Now you try!
4a) Find the sum of the measures of exterior angle of a
regular dodecagon.
4b) Find the value of r in polygon JKLM.
4r°
J
7r°
K
M
8r°
L
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5r°
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Photography Application
The appearance of the camera is formed by ten blades.
The blades overlap to form a regular decagon. What is
the measure of ∕CBD?
∕CBD is an exterior angle of a regular
decagon. By the polygon exterior angle
sum theorem, the sum of the exterior
measures is 360°.
m ∕CBD = 360° =36°
10
A
B
C
D
A regular decagon has 10
congruent ext. angles. So,
divide the sum by 10.
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Now you try!
5) Suppose the shutter of the camera were formed by
8 blades. What would the measure of each exterior
angle be?
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Now some problems for you to practice !
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Assessment
Tell whether each figure is a polygon. If it is a
polygon, name it by the number of its sides:
1)
2)
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Tell whether each polygon is regular or irregular.
Tell whether it is concave or convex:
3)
4)
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5) Find the measure of each interior angle of
pentagon ABCDE.
C
5x°
B 4x°
D
3x°
A
3x°
5x° E
6) Find the measure of each interior angle of a
regular dodecagon.
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7) Find the value of y in polygon JKLM.
J
4y°
K
2y°
4y°
6y°
M
L
8) Find the measure of each exterior angle of a regular
pentagon.
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9) Name the polygon by the number of its sides.
R
Q
S
P
T
10) In the polygon, /P, /R and /T are right angles and /Q is
congruent to /S. What are m/Q and m/S?
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Let’s review
Properties and Attributes of Polygons
Today you will learn about the parts of polygon and the ways
to classify polygons.
Each segment that forms a polygon is a side of the polygon.
The common endpoint of two sides is a vertex of the
polygon. A segment that connects any two nonconsecutive
vertices is a diagonal.
A
B
side
E
D
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C
vertex
diagonal
31
You can name a polygon by the number of its sides. The
table shows the names of some common polygons.
Polygon ABCDE in the previous slide is a pentagon.
Number of Slides
Name of Polygon
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
12
Dodecagon
n
n - gon
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Identifying Polygon
Tell whether each figure is a polygon. If it is a
polygon, name it by the number of its sides:
Polygon
Pentagon
Not a Polygon
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Polygon
Octagon
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All the sides are congruent in an equilateral polygon. All the
angles are congruent in an equiangular polygon. A regular
polygon is one that is both equilateral and equiangular. If a
polygon is not regular, it is called irregular.
A polygon is concave if any part of a diagonal contains points
in the exterior of the polygon. If no diagonal contains points in
the exterior, then the polygon is convex.
concave
quadrilateral
convex
quadrilateral
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Polygon Angle Sum Theorem
The sum of the interior angle measures of a convex
polygon with sides n is
(n - 2) 180°.
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Finding Interior Angle Measures
and Sums in Polygons
B) Find the measure of each interior angle of a regular nonagon.
Step1: Find the sum of the interior angle measures.
(n - 2) 180°
Polygon ∕ Sum thm.
= (9 - 2) 180°
Substitute 9 for n.
= 1260°
Simplify.
Step2: Find the measure of one interior angle.
1260° = 140°
9
The int. ∕s are congruent, so divide by 9.
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In the polygons below, an exterior angle has been
measured at each vertex. Notice that in each case, the
sum of the exterior angle measure is 360°.
81°
147°
132°
41°
147° + 81° + 132° = 360°
111°
43°
55°
110°
43° + 111° + 41° + 55°
+ 110° = 360°
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Polygon Exterior Angle Sum
Theorem
The sum of the exterior angle measures , one angle at
each vertex, of a convex polygon with sides n is
360°.
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Photography Application
The appearance of the camera is formed by ten blades.
The blades overlap to form a regular decagon. What is
the measure of ∕CBD?
∕CBD is an exterior angle of a regular
decagon. By the polygon exterior angle
sum theorem, the sum of the exterior
measures is 360°.
m ∕CBD = 360° =36°
10
A
B
C
D
A regular decagon has 10
congruent ext. angles. So,
divide the sum by 10.
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You did a great job today!
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