Quadrilaterals Chapter 6

Objectives Recognize and name convex polygons and regular polygons. Find the measures of interior angles and exterior angles of convex polygons.

Definition: Polygon A closed geometric figure in a plane formed by connecting line segments endpoint to endpoint with each segment intersecting exactly two others.

Definition: Convex Polygon A convex polygon is a simple polygon whose interior is a convex set. The following properties of a simple polygon are all equivalent to convexity: Every internal angle is less than 180 degrees or equal to 180 degrees. Every line segment between two vertices of the polygon does not go exterior to the polygon (i.e., it remains inside or on the boundary of the polygon).

Classifying Polygons

Classifying Polygons Polygons are also classified according to the number of sides. A triangle is a three-sided polygon. A quadrilateral is a four-sided polygon. A pentagon is a five-sided polygon. A hexagon is a six-sided polygon. A heptagon is a seven-sided polygon. An octagon is an eight-sided polygon. A nonagon is a nine-sided polygon. A decagon is a ten-sided polygon.

Classifying Polygons Regular Polygons When a polygon is both equilateral and equiangular, it is referred to as a regular polygon. For a polygon to be regular, it must also be convex.

Definition: Diagonal A diagonal of a polygon is any segment that joins two nonconsecutive vertices. Segments QS , SU , UR , RT and QT are the diagonals in this polygon.

Angle Measures in a Polygon Let n = the number of sides of the polygon then, the polygon has n vertices the polygon has n pairs of supplementary angles the sum of the measures of all the angles = 180n

Angle Measures in a Polygon If n = the number of sides of the polygon, then 180n = (interior angle sum) + (exterior angle sum); 180n = 180(n-2) + (exterior angle sum); 180n = 180n – 360 + (exterior angle sum); 360 = exterior angle sum.

Theorem 3-13 The sum of the measures of the angles of a convex polygon with n sides is (n - 2)180º.

Theorem 3-14 The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360º.

Review Quiz Find the measure of an interior angle: For a regular quadrilateral. For a regular octagon. For a regular polygon with 15 sides. For a regular polygon with 20 sides.

Review Quiz - Solutions Find the measure of an interior angle: For a regular quadrilateral. 90º For a regular octagon. 135º For a regular polygon with 15 sides. 156º For a regular polygon with 20 sides. 162º