POLYGONS

Examples of Polygons: NOT Examples of Polygons: Definition of a Polygon A polygon is a closed figure formed by a finite number of coplanar segments such that 1.the sides that have a common endpoint are noncollinear, and 2.each side intersects exactly two other sides, but only at their endpoints. not closed this side intersects 3 other sides this is not a side

interior of the polygon exterior of the polygon polygon Definition of a Convex/Concave Polygon A convex polygon is a polygon such that no line containing a side of a polygon contains a point in the interior of the polygon. A polygon that is not convex is nonconvex or concave. CONVEX POLYGONS CONCAVE POLYGONS

Polygons may be classified by the number of sides they have: Number of Sides Polygon triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon dodecagon

D E F G In the figure, what are the diagonals of quadrilateral DEFG? DF and EG Definition of a Diagonal of a Polygon A diagonal of a polygon is a segment that joins two nonconsecutive vertices of that polygon.

Consider each convex polygon below. From ONE vertex, draw all possible diagonals in each convex polygon. Then determine the number of triangles formed in each convex polygon after the diagonals are drawn from one vertex. quadrilateralpentagonhexagonheptagonoctagon Number of Triangles Formed =5 - 2 =6 - 2 =7 - 2 =8 - 2 =

PolygonNumber of Number of triangles formedSum of All Sides Angle Measures – 2 = 1 4 – 2 = 2 5 – 2 = 3 8 – 2 = 6 1 · 180 = · 180 = · 180 = · 180 = 1080

Theorem: The sum of the measures of the interior angles of a convex polygon with n sides is (n - 2)180. Ex: Find the measure of each interior angle in quadrilateral RSTU: S R T U 2x° x° 3x° 4x° Sol: 360 = m ∠ R + m ∠ S + m ∠ T + m ∠ U 360 = 2x + 4x + x + 3x 360 = 10x 36 = x m ∠ R = 72, m ∠ S = 144, m ∠ T = 36, m ∠ U = 108

Definition of a Regular Polygon A regular polygon is a convex polygon whose sides are all congruent and angles are also all congruent. Examples of Regular Polygons:

NOTE: Each angle of a regular convex polygon with n sides has a measure (n - 2)180. n Ex. Find the measure of one interior angle of a a) regular hexagonb) regular decagon a)(6 - 2) 180 = 120b) (10 – 2) 180 =

Theorem: The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.