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# The Polygon Angle-Sum Theorems

The Polygon Angle-Sum Theorems. 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.

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## The Polygon Angle-Sum Theorems

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1. The Polygon Angle-Sum Theorems

2. 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

3. 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

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

5. Classifying a Polygon by the number of sides:

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

7. Classify • Classify each polygon by its sides. Identify each as convex or concave:

8. 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:

9. Sum of Polygon Angle Measures

10. 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.

11. 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 :

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

13. 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:

14. 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°

15. 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°

16. 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.

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