section 3-4 polygon angle-sumtheorems name polygons calculate interior angles calculate exterior...
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Section 3-4Polygon Angle-SumTheorems
•Name Polygons
•Calculate Interior Angles
•Calculate Exterior Angles
•Understand Diagonals
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The Basics . . . Definition: A polygon is a closed
plane figure with at least three sides that are segments. The sides intersect only at their endpoints, and no adjacent sides are collinear.
PolygonsNot Polygons
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Naming Polygons Start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction.
D
I
AN
EDIANEIANEDANEDINEDIAEDIAN
DENAIENAIDNAIDEAIDENIDENA
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Definition: A diagonal of a polygon is a segment that connects two nonconsecutive vertices.
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The Basics . . . Definition: A convex polygon has all
diagonals on the interior of the polygon.
Definition: A concave polygon has a diagonal on the exterior of the polygon.
Convex Concave
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Naming Polygons
# Sides Polygon34567891012n
trianglequadrilateralpentagonhexagonheptagonoctagonnonagondecagondodecagonn-gon
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Real-Life Connections
Benzene
C C
C C
C C
H
HH
H
HH
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Apply What You Already Know:Theorem: The sum of the measures
of the three angles in a triangle is 180 degrees.
How About a Quadrilateral? . . . What is the sum of the angles? Explain your answer?
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Polygons and Interior AnglesInvestigation Part ITheorem: The sum Si of the measures
of the angles of a polygon with n sides is given by the formula: Si = (n – 2)180.
Huh?• The polygon is divided into triangles.• The number of triangles is always two
less than the number of polygon sides.• There are 180 in each triangle.
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Polygons and Exterior AnglesInvestigation Part II
Huh?
If you cut out each exterior angle and arranged the vertices on top of one another they would form a circle of sorts.
exterior angle
Theorem: If one exterior angle is taken at each vertex, the sum Se of the measures of the exterior angles of a polygon is given by the formula Se = 360
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More About Interior AnglesInvestigation Part III
180n – 360 = 180(n – 2)
Huh?
If you add all the angles of each triangle, then you include all the angles that go completely around the selected point. Hence, you need to subtract 360!
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Diagonals of PolygonsInvestigation Part III
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)3(
nnd
Theorem: The number, d, of diagonals that can be drawn in a polygon of n sides is given by the formula:Huh?From each of the n vertices you can draw n – 3 diagonals. Thus, there are n(n-3) diagonals total. But, by this method, each diagonal is drawn twice, so you must divide by 2.
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Regular PolygonsDefinition: A regular polygon is a polygon that is both equilateral and equiangular.
Theorem: The measure E of each exterior angle of an equiangular polygon of n sides is given by the formula
exterior angleE? find youcan how so
360,S that know already You e
nE
360