3.4 The Polygon Angle-Sum Theorems Chapter 3: Parallel and Perpendicular Lines

3.4 The Polygon Angle-Sum Theorems Polygon: a closed plane figure with at least three sides that are segments Not a polygon; Not enclosed Not a polygon; Two sides intersect A polygon

Naming a Polygon Name a polygon by its vertices. A ABCDE or AEDCB Start at one vertex and go around in order B E C D

Naming a Polygon Three polygons are pictured. Name each polygon: L P M

Classifying a Polygon by the number of sides: Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon

Convex vs. Concave A Convex Polygon has all vertices pointing “out” A Concave Polygon has one or more vertices “caving in”

Classify Classify each polygon by its sides. Identify each as convex or concave: Hexagon; Convex Octagon; Concave

Sum of Polygon Angle Measures Use triangles to figure out the sum of the angles in each polygon: # of Sides: # of Triangles: Total Degrees: # of Sides: # of Triangles: Total Degrees:

Sum of Polygon Angle Measures Number of Sides Number of Triangles Total Degrees inside Polygon 3 1 180 4 5 6 n

Theorem 3-9 Polygon Angle Sum Theorem The sum of the measures of the angles in a polygon is (n – 2)180. Find the sum of the measure of the angles of a 15-gon.

Polygon Angle Sum The sum of the measures of the angles of a given polygon is 720. How many sides does the polygon have? Use (n – 2)180 :

Using Polygon Angle-Sum Theorem Find the measure of <Y in pentagon TVYMR at the right. R T 135° Use (n – 2)180 M 90° Y V Write an equation to solve for <Y

Using Polygon Angle-Sum Theorem Pentagon ABCDE has 5 congruent angles. Find the measure of each angle. Use the Polygon Angle-Sum Theorem: (n – 2)180 Divide the total number of degrees by the number of angles:

Exterior Angles What do you notice about each set of exterior angles? 80° 75° 115° 2 1 150° 99° 130° 71° 70° 88° 1: 86° 3 2: 3: 70° 46°

Theorem 3-10 Polygon Angle-Sum Theorem The sum of one set of exterior angles for any polygon is 360°. 1 5 2 4 3 m<1 + m<2 + m<3 + m<4 + m<5 = 360°

Polygons Equilateral Polygon: all sides congruent Equiangular Polygon: all angles congruent Regular Polygon: all sides and all angles congruent (equiangular and equilateral) *If a polygon is a regular polygon then all of the exterior angles are also congruent.