Section 3-4Polygon Angle-SumTheorems

•Name Polygons

•Calculate Interior Angles

•Calculate Exterior Angles

•Understand Diagonals

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

Naming Polygons Start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction.

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Definition: A diagonal of a polygon is a segment that connects two nonconsecutive vertices.

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

Naming Polygons

# Sides Polygon34567891012n

trianglequadrilateralpentagonhexagonheptagonoctagonnonagondecagondodecagonn-gon

Real-Life Connections

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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?

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.

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

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!

Diagonals of PolygonsInvestigation Part III

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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.

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

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