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Find and use the measures of interior and exterior angles of polygons.

Objectives. Find and use the measures of interior and exterior angles of polygons. Remember!. By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.

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Find and use the measures of interior and exterior angles of polygons.

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  1. Objectives Find and use the measures of interior and exterior angles of polygons.

  2. Remember! By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.

  3. To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon.

  4. Example 3A: Finding Interior Angle Measures and Sums in Polygons Find the sum of the interior angle measures of a convex heptagon. (n – 2)180° Polygon  Sum Thm. (7 – 2)180° A heptagon has 7 sides, so substitute 7 for n. 900° Simplify.

  5. Example 3B: Finding Interior Angle Measures and Sums in Polygons 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° Polygon  Sum Thm. Substitute 16 for n and simplify. (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle. The int. s are , so divide by 16.

  6. Example 3C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. Polygon  Sum Thm. (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= 540 Combine like terms. c= 4 Divide both sides by 135.

  7. Example 3C Continued mA = 35(4°)= 140° mB = mE = 18(4°)= 72° mC = mD = 32(4°)= 128°

  8. Check It Out! Example 3a Find the sum of the interior angle measures of a convex 15-gon. (n – 2)180° Polygon  Sum Thm. (15 – 2)180° A 15-gon has 15 sides, so substitute 15 for n. 2340° Simplify.

  9. Check It Out! Example 3b Find the measure of each interior angle of a regular decagon. Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon  Sum Thm. Substitute 10 for n and simplify. (10 – 2)180° = 1440° Step 2 Find the measure of one interior angle. The int. s are , so divide by 10.

  10. 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°.

  11. Remember! An exterior angle is formed by one side of a polygon and the extension of a consecutive side.

  12. measure of one ext.  = Example 4A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each exterior angle of a regular 20-gon. A 20-gon has 20 sides and 20 vertices. sum of ext. s = 360°. Polygon  Sum Thm. A regular 20-gon has 20  ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20-gon is 18°.

  13. Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. Polygon Ext.  Sum Thm. 15b° + 18b° + 33b° + 16b° + 10b° + 28b°= 360° 120b= 360 Combine like terms. b= 3 Divide both sides by 120.

  14. measure of one ext. Check It Out! Example 4a Find the measure of each exterior angle of a regular dodecagon. A dodecagon has 12 sides and 12 vertices. sum of ext. s = 360°. Polygon  Sum Thm. A regular dodecagon has 12  ext. s, so divide the sum by 12. The measure of each exterior angle of a regular dodecagon is 30°.

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

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