Chapter 6

1. Chapter 6 6-1 properties of polygons

2. Objectives • Classify polygons based on their sides and angles. • Find and use the measures of interior and exterior angles of polygons.

3. Polygons • In previous lessons, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about ways to classify polygons.

4. Polygons • Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal.

5. Name of a polygon • You can name a polygon by the number of its sides. The table shows the names of some common polygons.

6. Remember • What is polygon? • A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.

7. Example 1A: Identifying Polygons • Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon

8. Example #2 • Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, heptagon

9. Example#3 • Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. not a polygon

10. Regular Polygon • All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygonis one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.

11. Classification • A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex.

12. Example • Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex

13. Example • Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave

14. Example • Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex

15. Find the sum of the interior angles • To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon.

16. Classification

17. Sum of the interior angles • In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles • is (n — 2)180°.

18. Example : Finding Interior Angle Measures and Sums in Polygons • Find the sum of the interior angle measures of a convex heptagon.

19. Example : Finding Interior Angle Measures and Sums in Polygons • Find the measure of each interior angle of a regular 16-gon.

20. Example : Finding Interior Angle Measures and Sums in Polygons • Find the measure of each interior angle of pentagon ABCDE.

21. The sum of the exterior angles • In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°.

22. Exterior Angle Sum Theorem

23. Example • Find the measure of each exterior angle of a regular 20-gon.

24. Example • Find the value of b in polygon FGHJKL.

25. Application • Ann is making paper stars for party decorations. What is the measure of 1? • 1 is an exterior angle of a regular pentagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°.

26. Student Guided Practice • Do even problems 2-13 in your book page 398

27. Homework • Do even problems from 16-26 in your book page 398 and 399

28. closure • Today we learned about properties of polygons • Next class we are going to learn about properties of parallelogram