1. 1 Polygons

2. 2 These figures are not polygonsThese figures are polygons Definition:A closed figure formed by line segments so that each segment intersects exactly two others, but only at their endpoints. Polygons

3. 3 Classifications of a Polygon Convex:No line containing a side of the polygon contains a point in its interior Concave: A polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon.

4. 4 Regular:A convex polygon in which all interior angles have the same measure and all sides are the same length Irregular: Two sides (or two interior angles) are not congruent. Classifications of a Polygon

5. Regular Polygons  Regular polygons have: All side lengths congruent All angles congruent 5

6. 6 Polygon Names 3 sides Triangle 4 sides 5 sides 6 sides 7 sides 8 sides Nonagon Octagon Heptagon Hexagon Pentagon Quadrilateral 10 sides 9 sides 12 sides Decagon Dodecagon n sides n-gon

7. 7 Measures of Interior and Exterior Angles  The name of a polygon depends on the number of sides in the polygon: triangle, quadrilateral, pentagon, hexagon, and so forth. The sum of the measures of the interior angles of a polygon also depends on the number of sides.

8. 8 Measures of Interior and Exterior Angles  For instance... Complete this table Polygon# of sides # of triangles Sum of measures of interior ’s Triangle 31 1●180=180 Quadrilateral 2●180=360 Pentagon Hexagon Nonagon (9) n-gon n

9. 9 Measures of Interior and Exterior Angles  What is the pattern? You may have found in the activity that the sum of the measures of the interior angles of a convex, n-gon is (n – 2) ● 180.  This relationship can be used to find the measure of each interior angle in a regular n-gon because the angles are all congruent.

10. 10 Polygon Interior Angles Theorem  The sum of the measures of the interior angles of a convex n-gon is (n – 2) ● 180  COROLLARY: The measure of each interior angle of a regular n-gon is: ● (n-2) ● 180 or

11. 11 Ex. 1: Finding measures of Interior Angles of Polygons  Find the value of x in the diagram shown: 88 136 142 105 xx

12. 12 SOLUTION:  The sum of the measures of the interior angles of any hexagon is (6 – 2) ● 180 = 4 ● 180 = 720.  Add the measure of each of the interior angles of the hexagon. 88 136 142 105 xx

13. 13 SOLUTION: 136 + 136 + 88 + 142 + 105 +x = 720. 607 + x = 720 X = 113 The sum is 720 Simplify. Subtract 607 from each side. The measure of the sixth interior angle of the hexagon is 113.

14. 14 Ex. 2: Finding the Number of Sides of a Polygon  The measure of each interior angle is 140. How many sides does the polygon have?  USE THE COROLLARY

15. 15 Solution: = 140 (n – 2) ●180= 140n 180n – 360 = 140n 40n = 360 n = 90 Corollary to Thm. 11.1 Multiply each side by n. Distributive Property Addition/subtraction props. Divide each side by 40.

16. 16 Notes  The diagrams on the next slide show that the sum of the measures of the exterior angles of any convex polygon is 360. You can also find the measure of each exterior angle of a REGULAR polygon.

17. 17 Copy the item below.

18. 18 EXTERIOR ANGLE THEOREMS

19. 19 Ex. 3: Finding the Measure of an Exterior Angle

20. 20 Ex. 3: Finding the Measure of an Exterior Angle

21. 21 Ex. 3: Finding the Measure of an Exterior Angle