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**Polygons Keystone Geometry**

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Polygons Definition: A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints. These figures are not polygons These figures are polygons

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**Classifications of a Polygon**

Convex: No line containing a side of the polygon contains a point in its interior Concave: A polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon.

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**Classifications of a Polygon**

Regular: A convex polygon in which all interior angles have the same measure and all sides are the same length Irregular: Two sides (or two interior angles) are not congruent.

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**Polygon Names Classified by the number of sides**

Triangle 4 sides Quadrilateral 5 sides Pentagon 6 sides Hexagon 7 sides Heptagon 8 sides Octagon 9 sides Nonagon 10 sides Decagon Dodecagon 12 sides n sides n-gon

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**Convex Polygon Formulas…..**

Diagonals of a Polygon: A segment connecting nonconsecutive vertices of a polygon For a convex polygon with n sides: The more sides a polygon has, the more diagonals it will have. A square has 2 diagonals. A pentagon has 5 diagonals. A hexagon has 9 diagonals………

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**Examples Diagonals in a square, n = 4 Diagonals in an octagon, n = 8**

** What about a triangle? A triangle does not have ANY diagonals. Diagonals in an octagon, n = 8

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**Polygon Angle Formulas**

Finding the INTERIOR angles in a convex polygon with n sides: The sum of the interior angles is The measure of one interior angle is Finding the EXTERIOR angles in a convex polygon with n sides: The sum of the exterior angles is The measure of one exterior angle is

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**Solving for the Angles Polygon Triangle Quadrilateral Pentagon Hexagon**

Interior Angle Sum 180 360 540 720 Each Interior Angle 60 90 108 120 Exterior Angle Sum Each Exterior Angle 72

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**Solving for the Angles Polygon Heptagon Octagon Nonagon Decagon**

Interior Angle Sum Each Interior Angle Exterior Angle Sum Each Exterior Angle

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**Examples….. Sum of the measures of the interior angles of a 11-gon is**

The measure of an exterior angle of a regular octagon is The number of sides of regular polygon with exterior angle 72 ° is The measure of an interior angle of a regular polygon with 30 sides

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