1. Essential Question – How can I find angle measures in polygons without using a protractor?

2. Polygons A polygon is a closed figure formed by a finite number of segments such that: 1. the sides that have a common endpoint are noncollinear, and 2. each side intersects exactly two other sides, but only at their endpoints.

3. Nonexamples

4. Polygons Can be concave or convex. Concave Convex

5. Polygons are named by number of sides Number of SidesPolygon 3 4 5 6 7 8 9 10 12 n Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon

6. Regular Polygon A convex polygon in which all the sides are congruent and all the angles are congruent is called a regular polygon.

7. Polygon# of sides# of trianglesSum of interior angles Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon n-gon 3 4 5 6 7 8 n 3 4 5 6 n - 2 2 1180° 2 · 180 = 360° 3 · 180 = 540° 4 · 180 = 720° 5 · 180 = 900° 6 · 180 = 1080° (n – 2) · 180°

8. Polygon Angle-Sum Theorem The sum of the measures of the angles of an n-gon is (n - 2) 180. A decagon has 10 sides, so n = 10. Sum = (n – 2)(180) Polygon Angle-Sum Theorem = (10 – 2)(180) Substitute 10 for n. = 8 180 Simplify. = 1440 Find the sum of the measures of the angles of a decagon.

9. Polygon Angle-Sum Theorem The sum of the measures of the angles of a given polygon is 720. How can you use the Polygon Angle- Sum Theorem to find the number of sides in the polygon? Sum = (n – 2) 180 Write the Equation 720 = (n – 2) 180 Sub. In known values 720 = 180n – 360 Simplify 1080 = 180n Addition Prop of EQ 6 = n Hexagon (6 sides)

10. Corollary to the Polygon Angle-Sum Theorem The measure of each interior angle of a regular n- gon is

11. How many degrees are in each individual interior angle of a… Hexagon Octagon Decagon

12. The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360 °. Each exterior angle of a regular polygon is 360 n where n is the number of sides in the polygon Polygon Exterior Angles Theorem

13. How many degrees are in each exterior angle of a … Hexagon Decagon Pentagon

14. 54⁰ 68⁰ 65⁰ (3x + 13)⁰ 60⁰ (4x – 12)⁰ Find the value for x. Sum of exterior angles is 360° (4x – 12) + 60+ (3x + 13) + 65 + 54+ 68 = 360 7x + 248 = 360 – 248 – 248 7x = 112 7 7 x = 12 Example What is the sum of the exterior angles in an octagon? What is the measure of each exterior angle in a regular octagon? 360° 360°/8= 45°

15. Polygon Interior Angles Theorem The sum of the measures of the interior angles of a convex n-gon is (n – 2) 180. Examples – 1. Find the sum of the measures of the interior angles of a 16–gon. 2. If the sum of the measures of the interior angles of a convex polygon is 3600 °, how many sides does the polygon have. 3. Solve for x. 4x - 2 82 108 2x + 10 (16 – 2)*180 (n – 2)*180 = 3600 180n – 360 = 3600 + 360 + 360 180n = 3960 180 180 n = 22 sides (4 – 2)*180 = 360 108 + 82 + 4x – 2 + 2x + 10 = 360 6x + 198 = 360 6x = 162 6 6 x = 27 = 2520°