1. Ch 3.5 Standard 12.0: Students find and use measures of interior and exterior angles of triangles to classify figures and solve problems. Standard 13.0 Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. Objective: 1)To find the sums of the measures of the interior and exterior angles of polygons.

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

3. Theorem http://www.phschool.com/atschool/academy123/english/ac ademy123_content/wl-book-demo/ph-385s.html

4. Example 1 Find the value of b in polygon FGHJKL. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° Polygon Ext.  Sum Thm. 120b = 360Combine like terms. b = 3Divide both sides by 120.

5. Example 2 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° 35c + 18c + 32c + 32c + 18c = 540 Substitute. 135c = 540Combine like terms. c = 4Divide both sides by 135. m  A = 35(4°) = 140° m  B = m  E = 18(4°) = 72° m  C = m  D = 32(4°) = 128°

6. Theorem 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°. http://www.phschool.com/atschool/academy123/english/acad emy123_content/wl-book-demo/ph-386s.html

7. Example 3 Find the value of r in polygon JKLM. 4r° + 7r° + 5r° + 8r° = 360° Polygon Ext.  Sum Thm. 24r = 360Combine like terms. r = 15Divide both sides by 24.

8. Example 4 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°. A regular pentagon has 5  ext. , so divide the sum by 5.