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CONFIDENTIAL 1
Geometry
Properties and Attributes of Polygons
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CONFIDENTIAL 2
Warm up
Solve by factoring:
1) x2 + 3x – 10 = 0
2) x2 - x – 12 = 0
3) x2 - 12x = - 35
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CONFIDENTIAL 3
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.
Properties and Attributes of Polygons
diagonal
A
E
B
CD
sidevertex
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CONFIDENTIAL 4
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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CONFIDENTIAL 5
Identifying Polygon
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:
PolygonPentagon
Not a PolygonPolygonOctagon
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CONFIDENTIAL 6
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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CONFIDENTIAL 7
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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CONFIDENTIAL 8
Classifying Polygons
Next page ->
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:
A
irregular convex
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CONFIDENTIAL 9
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:
B
regular convex
C
irregular concave
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CONFIDENTIAL 10
Now you try!
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:
2a 2b
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CONFIDENTIAL 11
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.
Triangle
Quadrilateral
Pentagon Hexagon
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CONFIDENTIAL 12
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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CONFIDENTIAL 13
The sum of the interior angle measures of a convex polygon with sides n is
(n - 2) 180°.
Polygon Angle Sum Theorem
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CONFIDENTIAL 14
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°
An octagon has 8 sides. So, substitute 8 for n.
Polygon ∕ Sum thm.
Simplify.
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CONFIDENTIAL 15
Finding Interior Angle Measures and Sums in Polygons
B) Find the measure of each interior angle of a regular nonagon.
(n - 2) 180°
= (9 - 2) 180°
= 1260°
Substitute 9 for n.
Polygon ∕ Sum thm.
Simplify.
Step1: Find the sum of the interior angle measures.
Step2: Find the measure of one interior angle.
1260° = 140° 9
The int. ∕s are congruent, so divide by 9.
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CONFIDENTIAL 16
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° => c = 45°
Polygon ∕ Sum thm.
Polygon ∕ Sum thm.
Substitute.
P
Q R
Sc°
3c° c°
3c°
m∕P =m∕R = 45°
m∕Q = m∕S = 360°
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CONFIDENTIAL 17
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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CONFIDENTIAL 18
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°
132°147°
147° + 81° + 132° = 360°
41°55°
110°43°
111°
43° + 111° + 41° + 55° + 110° = 360°
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CONFIDENTIAL 19
The sum of the exterior angle measures , one angle at each vertex, of a convex polygon with sides n is
360°.
Polygon Exterior Angle Sum Theorem
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CONFIDENTIAL 20
Finding Exterior Angle Measures in Polygons
A) Find the measure of each exterior angle of a regular hexagon.
A regular hexagon has 6 ext ∕s. So, divide the sum by 6.
Polygon ext ∕ Sum thm.
A hexagon has 6 sides and 6 vertices.
Sum of the exterior angle = 360°
Measure of one exterior angle = 360° = = 60° 6
The measure of each exterior angle of a regular hexagon = 60°
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CONFIDENTIAL 21
Finding Exterior Angle Measures in Polygons
B) Find the value of a in polygon RSTUV.
Combine like terms.
Polygon ext ∕ Sum thm. 7a° + 2a° + 3a° + 6a° + 2a° = 360°
20a°= 360°
a = 60°
2a°3a°
6a°2a°
7a°
R
ST
U
V
So, divide the sum by 20.
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CONFIDENTIAL 22
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.
J
K
L
M
4r°7r°
5r°8r°
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CONFIDENTIAL 23
Photography ApplicationThe appearance of the camera is formed by ten blades. The blades overlap to form a regular decagon. What is
the measure of ∕CBD?
m ∕CBD = 360° =36° 10
A regular decagon has 10 congruent ext. angles. So, divide the sum by 10.
∕CBD is an exterior angle of a regular decagon. By the polygon exterior angle sum theorem, the sum of the exterior
measures is 360°. B
D
A
C
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CONFIDENTIAL 24
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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CONFIDENTIAL 25
Now some problems for you to practice !
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CONFIDENTIAL 26
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:
Assessment
1) 2)
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CONFIDENTIAL 27
3) 4)
Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:
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CONFIDENTIAL 28
5) Find the measure of each interior angle of pentagon ABCDE.
A
B
C
D
E3x°
3x°
4x°
5x°
5x°
6) Find the measure of each interior angle of a regular dodecagon.
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CONFIDENTIAL 29
7) Find the value of y in polygon JKLM.
8) Find the measure of each exterior angle of a regular pentagon.
J
K
L
M
4y°
4y° 2y°
6y°
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CONFIDENTIAL 30
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
P
Q S
T
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CONFIDENTIAL 31
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.
Properties and Attributes of Polygons
diagonal
A
E
B
CD
sidevertex
Let’s review
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CONFIDENTIAL 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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CONFIDENTIAL 33
Identifying Polygon
Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:
PolygonPentagon
Not a PolygonPolygonOctagon
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CONFIDENTIAL 34
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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CONFIDENTIAL 35
The sum of the interior angle measures of a convex polygon with sides n is
(n - 2) 180°.
Polygon Angle Sum Theorem
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CONFIDENTIAL 36
Finding Interior Angle Measures and Sums in Polygons
B) Find the measure of each interior angle of a regular nonagon.
(n - 2) 180°
= (9 - 2) 180°
= 1260°
Substitute 9 for n.
Polygon ∕ Sum thm.
Simplify.
Step1: Find the sum of the interior angle measures.
Step2: Find the measure of one interior angle.
1260° = 140° 9
The int. ∕s are congruent, so divide by 9.
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CONFIDENTIAL 37
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°
132°147°
147° + 81° + 132° = 360°
41°55°
110°43°
111°
43° + 111° + 41° + 55° + 110° = 360°
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CONFIDENTIAL 38
The sum of the exterior angle measures , one angle at each vertex, of a convex polygon with sides n is
360°.
Polygon Exterior Angle Sum Theorem
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CONFIDENTIAL 39
Photography ApplicationThe appearance of the camera is formed by ten blades. The blades overlap to form a regular decagon. What is
the measure of ∕CBD?
m ∕CBD = 360° =36° 10
A regular decagon has 10 congruent ext. angles. So, divide the sum by 10.
∕CBD is an exterior angle of a regular decagon. By the polygon exterior angle sum theorem, the sum of the exterior
measures is 360°. B
D
A
C
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CONFIDENTIAL 40
You did a You did a great great job today!job today!