Math 2 Geometry Based on Elementary Geometry, 3 rd ed, by Alexander & Koeberlein 2.5 Convex Polygons.

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Presentation transcript:

Math 2 Geometry Based on Elementary Geometry, 3 rd ed, by Alexander & Koeberlein 2.5 Convex Polygons

Definition A polygon is a closed plane figure whose sides are line segments that intersect only at the endpoints. In this text only: convex polygons (interior angles measure from 0  to 180  ) Examples?

Polygon Examples Convex Polygons Concave Polygons

Not Polygons

Classify by number of sides Triangle(3) Quadrilateral (4) Pentagon (5)

Classify by number of sides Hexagon (6) Heptagon (7) Octagon (8)

Classify by number of sides Nonagon (9) Decagon (10)

Informal Definition A diagonal of a polygon is a line segment that joins two nonconsecutive vertices. A E D C B

Theorem The total number of diagonals D in a polygon of n sides is given by the formula:

Use the formula to find the number of diagonals of: a triangle a quadrilateral a pentagon

Theorem The sum S of the measures of the interior angles of a polygon with n sides is given by: S = (n – 2) 180 

Informal Definitions Equiangular polygon: All angles are congruent. Equilateral polygon: The measure of all sides are equal.

Definition A regular polygon is a polygon that is both equilateral and equiangular. Notice: A square is a regular quadrilateral.

Not Regular Polygons Notice it is possible for Polygons to be either equilateral or equiangular but not both. Rhombus Rectangle

Corollary The measure I of each interior angle of a regular polygon of n sides is:

Corollary The sum of the four interior angles of a quadrilateral is 360 

Corollary The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360 .

Corollary The measure E of each exterior angle of a regular polygon of n sides is:

Informal Definition A polygram is the star-shaped figure that results when the sides of certain polygons are extended. Pentagram Hexagram