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Polygon Properties: Interior and Exterior Angles Explained

Learn about polygons, interior and exterior angles, regular vs. irregular polygons, convex vs. concave shapes, and how to find angle measures. Practice evaluating expressions and solving for unknowns in triangles and quadrilaterals.

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Polygon Properties: Interior and Exterior Angles Explained

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  1. Warm Up 1.A ? is a three-sided polygon. 2. A ? is a four-sided polygon. Evaluate each expression for n = 6. 3. (n – 4) 12 4. (n – 3) 90 Solve for a. 5. 12a + 4a + 9a = 100 triangle quadrilateral 24 270 4

  2. Notes 8-1 Angles of Polygons

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

  4. A regular polygonis one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. Remember: 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.

  5. Diagonals divide a polygon into triangles Sum of Interior Angles Number of Triangles

  6. 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°.

  7. Example: Find the sum of the interior angle measures of a convex 15-gon. (n – 2)180° (15 – 2)180° 2340°

  8. Example: Find the measure of each interior angle of a regular 16-gon. Step 1 Find the sum of the interior angle measures. (n – 2)180° (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle.

  9. Example: Find the measure of each interior angle of a regular decagon. Step 1 Find the sum of the interior angle measures. (n – 2)180° (10 – 2)180° = 1440° Step 2 Find the measure of one interior angle.

  10. In the polygons below, an exterior angle has been measured at each vertex.

  11. Example Find the measure of each exterior angle of a regular 20-gon.

  12. Example: Find the measure of each exterior angle of a regular dodecagon. The measure of each exterior angle of a regular dodecagon is 30°.

  13. Example: Find the value of b in polygon FGHJKL. b= 3

  14. Example: Find the value of r in polygon JKLM. r= 15

  15. Lesson Quiz 1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex. 2. Find the sum of the interior angle measures of a convex 11-gon. nonagon; irregular; concave 1620° 3. Find the measure of each interior angle of a regular 18-gon. 4. Find the measure of each exterior angle of a regular 15-gon. 160° 24°

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