1. Section 6-1 Properties of Polygons

2. Classifying Polygons Polygon: Closed plane figure with at least three sides that are segments intersecting only at their endpoints, and no adjacent sides are collinear. A B C D E A B C D E A B C D E Not a polygon Polygon

3. Classifying Polygons Note: A diagonal of a polygon is a segments that connects two nonconsecutive vertices. Convex Polygon: A polygon with no diagonal with points outside the polygon Concave Polygon: A polygon with at least one diagonal with points outside the polygon. Convex Concave

4. Classifying Polygons – Example 1 Classify each polygon by its sides. Identify each as convex or concave. Convex Octogon Concave Hexagon Concave 20-gon

5. Classifying Polygons Equilateral Polygon: A polygon with all sides congruent. Equiangular Polygon: A polygon with all angles congruent. Regular Polygon: A polygon that is both equilateral and equiangular.

6. Classifying Polygons We can classify polygons according to the number of sides it has. Sides Name 3 4 5 6 8 9 10 12 n Triangle Quadrilateral Pentagon Hexagon Octagon Nonagon Decagon Dodecagon n-gon

7. Classifying Polygons – Example 2 Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. Regular, Convex Irregular; Concave Regular, Convex

8. Naming Polygons C D G H J K F Vertices: Sides: Angles: Polygon: DGHJKFC or GDCFKJH To name a polygon, start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction. C,D,G,H,J,K,F DG, GH, HJ, JK, KF, FC, CD <D, <G, <H, <J, <K, <F, <C

9. Class Work…  Complete the table with a partner/ group Theorem 3-14: Polygon Angle-Sum Theorem: The sum of the measures of the angles in an n-gon is (n – 2)180.

10. Assignment #39 Pg. 386 #2-8 all AND DEFINE TERMS: Polygon, Convex Polygon, Concave Polygon, Equilateral Polygon, Equiangular Polygon, Regular Polygon, & Polygon Angle Sum Theorem

11. Section 6-1 (cont.) Properties of Polygons

12. Theorem 3-14: Polygon Angle-Sum Theorem: The sum of the measures of the angles in an n-gon is (n – 2)180. Polygon Angle Sum Theorem

13. Polygon Angle Sum – Example 3 Find the sum of the measures of the angles of a regular dodecagon. Then find the measure of each interior angle. = (n – 2)180 = (12 – 2)180 = (10)180 = 1800 Polygon Angle Sum Theorem Substitution Simplify **Be Careful!! Are they asking for the SUM of the angles, or for EACH angle measure?**

14. The sum of the angles of a polygon is 720. Find the number of sides the polygon has and classify it. Sum of the Angles = (n – 2)180 720 = (n – 2)180 4 = (n – 2) 6 = n Hexagon Polygon Angle Sum Theorem Substitution Divide both sides by 180 Add 2 to both sides Example 4

15. Example 5 Find x in the following polygon. 125° x°x° To solve, we will use the polygon angle sum theorem for n=5 90 + 90 + 125 + x + 125 = (5 – 2)180 Polygon Angle Sum Theorem 330 + x = (5 - 2)180Simplify 330 + x = 540Simplify x = 210Subtract 330 from both sides

16. Example 6 Find the measure of each angle in the following polygon. To solve, we will use the polygon angle sum theorem for n=5 Polygon Angle Sum Theorem Simplify / Combine Like Terms Simplify Solve for x Now, solve for each angle in the polygon…

17. Polygon Angle Sums Theorem 3-15: Polygon Exterior Angle-Sum Theorem: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. For a pentagon: 1 2 3 4 5

18. Polygon Angle Sums – Example 7 Find the measure of each exterior angle of a regular dodecagon. **The sum of the exterior angles is 360 degrees** **Remember, a regular polygon has both equal sides and equal angles 360 = x(12) Set up an equation Now, Solve. 30 = x Divide both sides by 12 Each exterior angle is 30 degrees Exterior Angle Sum= Measure of Each Angle (Number of Sides)

19. Example 8 Find the measure of each angle in the following polygon. To solve, we will use the polygon exterior angle sum theorem & set all angles equal to 360 Polygon Exterior Angle Sum Theorem Simplify / Combine Like Terms Solve for b Now, solve for each angle in the polygon…

20. Assignment #40 Pg. 386 #9-15 all #16-21 all