Objectives. 6-1 Properties and Attributes of Polygons

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

2 side of a polygon vertex of a polygon diagonal regular polygon concave convex Vocabulary

3 In Lesson 2-4, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about ways to classify polygons.

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

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

7 Example 1: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon polygon, heptagon not a polygon

8 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.

9 A polygon is concaveif 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.

10 Example 2: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex irregular, concave regular, convex

11 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.

12 Remember! By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180.

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14 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.

15 Example 3a: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of a regular 16-gon. Step 1Find the sum of the interior angle measures. (n 2)180 (16 2)180 = 2520 Polygon Sum Thm. Substitute 16 for n and simplify. Step 2Find the measure of one interior angle. The int. s are, so divide by 16.

16 Example 3b: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. (5 2)180 = 540 Polygon Sum Thm. m A + m B + m C + m D + m E = 540 Polygon Sum Thm. 35c + 18c+ 32c+ 32c+ 18c= 540 Substitute. 135c= 540 Combine like terms. c= 4 Divide both sides by 135.

17 Example 3C Continued m A = 35(4 )= 140 m B = m E = 18(4 )= 72 m C = m D = 32(4 )= 128

18 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.

19 Remember! An exterior angle is formed by one side of a polygon and the extension of a consecutive side.

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21 Example 4A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each exterior angle of a regular 20-gon. A 20-gon has 20sides and 20vertices. sum of ext. s = 360. Polygon Sum Thm. measure of one ext. = A regular 20-gon has 20 ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20-gon is 18.

22 Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. Polygon Ext. Sum Thm. 15b + 18b + 33b + 16b + 10b + 28b = b= 360 Combine like terms. b= 3 Divide both sides by 120.

23 Example 5: Art Application Ann is making paper stars for party decorations. What is the measure of 1? 1is an exterior angle of a regular pentagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360. A regular pentagon has 5 ext., so divide the sum by 5.