2.7.1 properties of polygons
TRANSCRIPT
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2.7.1 Properties of Polygons
The student is able to (I can):
Name polygons based on their number of sides
Classify polygons based on
concave or convex
equilateral, equiangular, regular
Calculate and use the measures of interior and exterior angles of polygons
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polygon A closed plane figure formed by three or more noncollinear straight lines that intersect only at their endpoints.
polygons
notpolygons
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vertex
diagonal
regular
The common endpoint of two sides. Plural: verticesverticesverticesvertices.
A segment that connects any two nonconsecutive vertices.
A polygon that is both equilateral and equiangular.
vertexdiagonal
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Polygons are named by the number of their sides:
SidesSidesSidesSides NameNameNameName
3333 Triangle
4444 Quadrilateral
5555 Pentagon
6666 Hexagon
7777 Heptagon
8888 Octagon
9999 Nonagon
10101010 Decagon
12121212 Dodecagon
nnnn n-gon
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Examples Identify the general name of each polygon:
1.
2.
3.
pentagon
dodecagon
quadrilateral
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concave
convex
A diagonal of the polygon contains points outside the polygon. (caved in)
Not concave.
concave pentagon
convex quadrilateral
outside the polygon
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We know that the angles of a triangle add up to 180, but what about other polygons?
Draw a convex polygon of at least 4 sides:
Now, draw all possible diagonals from one vertex. How many triangles are there?
What is the sum of their angles?
180180180180
180180180180
180180180180
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Thm 6-1-1 Polygon Angle Sum Theorem
The sum of the interior angles of a convex polygon with n sides is
(n 2)180
If the polygon is equiangular, then the measure of one angle is
( 2)180n
n
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SidesSidesSidesSides NameNameNameName TrianglesTrianglesTrianglesTriangles Sum Int.Sum Int.Sum Int.Sum Int.Each Int.Each Int.Each Int.Each Int.(Regular)(Regular)(Regular)(Regular)
3333 Triangle 1 (1)180=180 60
4444 Quadrilateral 2 (2)180=360 90
5555 Pentagon 3 (3)180=540 108
6666 Hexagon
7777 Heptagon
8888 Octagon
9999 Nonagon
10101010 Decagon
12121212 Dodecagon
nnnn n-gon
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Lets update our table:
SidesSidesSidesSides NameNameNameName TrianglesTrianglesTrianglesTriangles Sum Int.Sum Int.Sum Int.Sum Int.Each Int.Each Int.Each Int.Each Int.(Regular)(Regular)(Regular)(Regular)
3333 Triangle 1 (1)180=180 60
4444 Quadrilateral 2 (2)180=360 90
5555 Pentagon 3 (3)180=540 108
6666 Hexagon 4 (4)180=720 120
7777 Heptagon 5 (5)180=900 128.6
8888 Octagon 6 (6)180=1080 135
9999 Nonagon 7 (7)180=1260 140
10101010 Decagon 8 (8)180=1440 144
12121212 Dodecagon 10 (10)180=1800 150
nnnn n-gon n 2 (n 2)180 ( 2)180n
n
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An exterior angle is an angle created by extending the side of a polygon:
Now, consider the exterior angles of a regular pentagon:
Exterior angle
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From our table, we know that each interior angles is 108. This means that each exterior angle is 180 108 = 72.
The sum of the exterior angles is therefore 5(72) = 360. It turns out this is true for any convex polygon, regular or not.
108
72
72
7272
72
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Polygon Exterior Angle Sum Theorem
The sum of the exterior angles of a convex polygon is 360.
For any equiangular convex polygon with n sides, each exterior angle is
360
n
SidesSidesSidesSides NameNameNameName Sum Ext.Sum Ext.Sum Ext.Sum Ext. Each Ext.Each Ext.Each Ext.Each Ext.
3333 Triangle 360 120
4444 Quadrilateral 360 90
5555 Pentagon 360 72
6666 Hexagon 360 60
8888 Octagon 360 45
nnnn n-gon 360 360/n