A of a polygon is a segment that joins two nonconsecutive vertices. 1. How many degrees are in a triangle?

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1 8.1- Find Angle Measures in Polygons SWBAT: find interior and exterior angle measures in polygons. Common Core: G.CO.11, G.CO.13, G.SRT.5 Do Now Fill in the blank. A of a polygon is a segment that joins two nonconsecutive vertices. Polygon Investigation 1. How many degrees are in a triangle? Interior Angles Sum The number of triangles that can be drawn within the polygon can determine the sum of the degrees in any polygon. Complete the table below and look for a pattern to find the sum of the degrees in any polygon. Polygon Picture # of Sides # of Triangles Sum of Interior s Triangle Quadrilateral

2 Pentagon Hexagon Heptagon X Octagon X Nonagon X Decagon X Sum of the Interior Angles of a Polygon If n represents the number of sides of a polygon, then the sum of the interior angles(s) can be found using the formula:

3 2. What does it mean for a polygon to be regular? Regular Polygon Sum of Interior Angles Measure of each Interior Angle Regular Polygon Sum of Interior Angles Measure of each Interior Angle Triangle Heptagon Quadrilateral Octagon Pentagon Nonagon Hexagon Decagon Measure of the Interior Angle of a Regular Polygon A regular polygon is one in which all the sides are equal; therefore all the are equal. To find the measure of an interior angle in a regular polygon use: Example 1: Interior Angles 1. Find the sum of the interior angles of a 15- gon. 2. Find the sum of interior angles of a 21- gon.

4 3. What is the measure of each interior angle of a regular dodecagon? 4. What is the measure of each interior angle of a regular 18- gon? 5. The measure of an interior angle of a regular polygon is 165, how many sides does this polygon have? 6. The measure of an interior angle of a regular polygon is 156, how many sides does this polygon have? Polygon Exterior Angles: The sum of the measures of the exterior angles of a convex polygon is

5 Finding the measure of exterior angle of a regular polygon. Divide 360 by the number of sides to get the measure of each exterior angle. Example 2: Exterior Angles 7. What is the measure of each exterior angle of a regular hexagon? 8. What is the measure of each exterior angle of a regular nonagon? 9. What is the measure of each exterior angle of a regular 24- gon? 10. What is the measure of each exterior angle of a regular 30- gon? 11. If the exterior angle of a regular polygon measures 12, how many sides does the polygon have? 12. If the exterior angle of a regular polygon measures 40, how many sides does the polygon have? 13. The measure of the exterior angles of a convex quadrilateral are 90, 10x, 5x, and 45. What is the measure of the largest exterior angle?

6 14. The measure of the interior angles of a convex octagon are 45x, 40x, 155, 120, 155, 38x, 158, and 41x. What is the measure of the smallest interior angle? Closure: Interior & Exterior Angles of a Polygon Ø Sum of Interior Angles of a Polygon: Ø Interior Angles of a Regular Polygon: Ø Sum of Exterior Angles of a Polygon: Ø Exterior Angles of a Regular Polygon: Ø Number of Sides of a Regular Polygon: Homework: 1. What is the sum of the measuresof the interior angles of a 27- gon? 2. What is the measure of each interior angle of a regular 20- gon? 3. The sum of the interior angles of a polygon is How many sides does the polygon have? 4. The measures of seven angles in a nonagon measure 138, 154, 145, 132, 128, 147, and 130. If the two remaining angles are equal in measure, what is the measure of each angle?

7 5. What is sum of the measures of the exterior angles of a heptagon? 6. What is the measure of each exterior angle of a 20- gon? 7. If the exterior angle of a regular polygon measures 9, how many sides does the polygon have? 8. If the interior angles of a regular polygon measure 108, how many sides does the polygon have? 9. Find the value of x. 10. Find the value of x. 11. Find the value of x.

8 12. ABCF EDCF. Find the value of x. 13. Find the value of x. 14. Find m BAC. 15. Find the value of x.

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