Geometry 3.5 Angles of a Polygon Standard 12.0 & 13.0

Polygons (“many angles”) have vertices, sides, angles, and exterior angles are named by listing consecutive vertices in order AB C D E F Hexagon ABCDEF

Polygons formed by line segments, no curves the segments enclose space each segment intersects two other segments

Not Polygons Polygons

Diagonal of a Polygon A segment connecting two nonconsecutive vertices Diagonals

Convex Polygons No side ”collapses” in toward the center Easy test : RUBBER BAND stretched around the figure would have the same shape…….

Nonconvex Polygons Convex Polygons

When the textbook refers to polygons, it means convex polygons From now on…….

Polygons are classified by number of sides Number of sidesName of Polygon 3triangle 4quadrilateral 5pentagon 6hexagon 8octagon 10decagon nn-gon

Interior Angles of a Polygon To find the sum of angle measures, divide the polygon into triangles Draw diagonals from just one vertex 4 sides, 2 triangles Angle sum = 2 (180) 5 sides, 3 triangles Angle sum = 3 (180) 6 sides, 4 triangles Angle sum = 4 (180) DO YOU SEE A PATTERN ?

Interior Angles of a Polygon 4 sides, 2 triangles Angle sum = 2 (180) 5 sides, 3 triangles Angle sum = 3 (180) 6 sides, 4 triangles Angle sum = 4 (180) The pattern is: ANGLE SUM = (Number of sides – 2) (180)

Theorem The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)180. Example: 5 sides. 3 triangles. Sum of angle measures is (5-2)(180) = 3(180) = 540

Exterior Angles of a Polygon Draw the exterior angles Put them together The sum = 360 Works with every polygon!

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

If a polygon is both equilateral and equiangular it is called a regular polygon Regular Polygons 120 Equilateral Equiangular 120 Regular

Example 1 A polygon has 8 sides (octagon.) Find: a) The interior angle sum b) The exterior angle sum n=8, so (8-2)180 = 6(180) =

Example 2 Find the measure of each interior and exterior angle of a regular pentagon Interior:(5-2)180 = 3(180) = = 108 each 5 Exterior:360 = 72 each 5

Example 3 How many sides does a regular polygon have if: a)the measure of each exterior angle is = = 45n n n = 88 sides: an octagon b) the measure of each interior angle is 150 (n-2)180 = 150(n-2)180 = 150n n180n – 360 = 150n = - 30n n = sides

Homework