Geometry

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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.
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 Octagon
Polygon Pentagon
Not a Polygon
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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
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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 -gt
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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 (8 - 2) 180 1080
Polygon / Sum thm.
An octagon has 8 sides. So, substitute 8 for n.
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 (9 - 2) 180 1260
Polygon / Sum thm.
Substitute 9 for n.
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.
(4 - 2) 180 360 m/P m/Q m/R m/S
360 c 3c c 3c 360 8c
360 gt c 45
Polygon / Sum thm.
Polygon / Sum thm.
Substitute.
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.
147 81 132 360
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 Measure of one
exterior angle 360 60 6
Polygon ext / Sum thm.
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 2a 3a 6a 2a 360
20a 360 a 60
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.
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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 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
4)
3)
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5) Find the measure of each interior angle of
pentagon ABCDE.
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.
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.
10) In the polygon, /P, /R and /T are
right angles and /Q is congruent to /S. What are
m/Q and m/S?
R
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Lets 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.
diagonal
32
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 Octagon
Polygon Pentagon
Not a Polygon
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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
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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 (9 - 2) 180 1260
Polygon / Sum thm.
Substitute 9 for n.
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.
147 81 132 360
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 regular decagon has 10 congruent ext. angles.
So, divide the sum by 10.
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You did a great job today!