Polygons and Their Angles

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Presentation transcript:

Polygons and Their Angles

Polygon: A closed figure Polygon: A closed figure. They have vertices, sides, angles, and exterior angles. You name a polygon by just listing its vertices in order around the polygon. ABCDEF One name for this polygon is _________________________ Diagonal: a segment connecting two NONconsecutive vertices of a polygon. AC or AD or AE Diagonal Example: __________________

Convex Polygons - polygons where no diagonal goes outside the figure. Concave Polygons - polygons where any diagonal goes outside of the figure. Concave polygons “cave” in.

Classifying Polygons You can classify a polygon by the number of sides it has. YOU WILL BE EXPECTED TO KNOW THESE!!

INTERIOR ANGLE SUM THEOREM The sum of the measures of the angles in a convex polygon with n sides is (n - 2)180

Exterior Angle Sum Theorem The sum of the measure of the exterior angles of ANY convex polygon, one at each vertex is 360

Example Find the interior and exterior angle sums for each polygon: 1. quadrilateral 2. 12-gon 3. hexagon 4. nonagon (4-2)180 = 360 5. decagon 6. pentagon 7. octagon 8. 18-gon (10-2)180 = 1440 Ext. Angle Sum = 360 Exterior Angle sum is always 360 (12-2)180 = 1800 Ext. Angle Sum = 360 (5-2)180 = 540 Ext. Angle sum = 360 (6-2)180 = 720 Ext. Angle Sum = 360 (8-2)180 = 1080 Ext Angle Sum = 360 (9-2)180 = 1260 Ext. Angle Sum = 360 (18-2)180 = 2880 Ext Angle Sum = 360

Example 2 Find the value of x You should get x = 100 Since there are 5 sides, then the interior angle sum is (5-2)180 or 540. Then take 360 - 90 - 90 - 160 - 150 to get x. You should get 50 Do the same for the others. Count the number of sides and figure the interior angle sum. Then subtract out the angles that you already know. You know there are 4 sides, so the interior angle sum is 360. Take 360 - 60 and you get 300. Then divide by 3 and each angle is 100.

Regular Polygons Regular Polygons - a polygon that is BOTH equilateral AND equiangular If you see the word REGULAR, it means the figure is special and you can divide by the number of sides to get individual angle measures

NOTICE: interior and exterior angles add to 180!!!! Example 3 For each REGULAR polygon, find the measure of each interior angle and exterior angle. 13. triangle 14. quadrilateral 15. hexagon Interior =(3-2)180 / 3 = 60 16. decagon 17. 15-gon Exterior = 360 / 3 = 120 Interior = (10-2)180 / 10 = 144 Exterior = 360 / 10 = 36 Interior = (4-2)180 / 4 = 90 Exterior = 360 / 4 = 90 Interior = (15-2)180 / 15 = 156 Exterior = 360 / 15 = 24 Interior = (6-2)180 / 6 = 120 Exterior = 360 / 6 = 60 NOTICE: interior and exterior angles add to 180!!!!

Example 4 How many sides does a regular polygon have if the measure of each exterior angle is: Just take 360 divided by each angle to get your answer 18. 60 19. 15 20. 120 6 sides 24 sides 3 sides

Example 5 How many sides does a regular polygon have if the measure of each interior angle is: Since the interior and exterior angles Add to 180, find the exterior angle first!!! Interior angle is 60, so exterior angle is 180-60 = 120. Now do 360 divided by 120. You should get 3. 21. 60 22. 160 23. 144 Interior angle is 160. 180-160 = 20, so exterior angle is 20. Now do 360 divided by 20. You get 18 Interior angle is 144. 180-144 = 36, so exterior angle is 36. Now do 360 divided by 36. You get 10.

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