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Exploring the Interior Angles of Polygons: Measuring Angles Effectively

In this section, we delve into the fascinating world of interior angles in polygons. Just as the sum of the interior angles of a triangle is 180°, other polygons have their own rules. For instance, the sum of interior angles can be calculated using the formula (180(n-2)), where (n) is the number of sides. This guide provides step-by-step instructions on finding the interior angle sums of various polygons, including heptagons and 30-gons. We'll also explore how to determine the number of sides from given angle sums and vice versa.

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Exploring the Interior Angles of Polygons: Measuring Angles Effectively

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  1. 2.6 What is Its Measure? Pg. 22 Interior Angles of a Polygon

  2. 2.6 – What is Its Measure?_____________ Interior Angles of a Polygon In an earlier chapter, you discovered that the sum of the interior angles of a triangle is always 180°. But what about the sum of the interior angles of other polygons, such as hexagons or decagons? Consider these questions today as you investigate the angles of all polygons.

  3. Angle inside a shape Angle outside a shape 1 2

  4. 1 180° 2 360° 3 540° 4 720° 5 900°

  5. 6 1080° 7 1260° 8 1440° n – 2 180(n – 2) n-gon

  6. 180(n – 2) n 180(n – 2)

  7. 1. Find the sum of the measures of the interior angles of the indicated polygon. 180(n – 2) 180(7 – 2) 180(5) 900° heptagon Name __________________ Polygon Sum = __________ 900°

  8. 1. Find the sum of the measures of the interior angles of the indicated polygon. 30-gon 180(n – 2) 180(30 – 2) 180(28) 5040° 30-gon Name __________________ Polygon Sum = __________ 5040°

  9. 2. Find the indicated variable. 180(n – 2) 1 180(5 – 2) 5 180(3) 540° 2 540° 4 x + 90 +143 + 77 + 103 = 540 3 x + 413 = 540 x = 127°

  10. 1 180(n – 2) 7 2 180(7 – 2) 6 900° 3 5 4 3m + 501 = 900 3m = 399 m = 133°

  11. 3. Given the sum of the measures of the interior angles of a polygon, find the number of sides. 2340° 180(n – 2) = 2340 180 180 ? n – 2 = 13 2340° n = 15

  12. 3. Given the sum of the measures of the interior angles of a polygon, find the number of sides. 6840° 180(n – 2) = 6840 180 180 ? n – 2 = 38 6840° n = 40

  13. 4. Given the number of sides of a regular polygon, find the measure of each interior angle. 180(n – 2) 180(5 – 2) = = n 5 540 180(3) = = 108° 5 5

  14. 18 sides 180(n – 2) 180(18 – 2) 180(16) = = = n 18 18 2880 = 160° 18

  15. 5. Given the measure of an interior angle of a regular polygon, find the number of sides. 144° 180(n – 2) 144 = n 1 180(n – 2) = 144n 180n – 360 = 144n 36n – 360 = 0 144° 144° 36n = 360 n = 10

  16. 108° 180(n – 2) 108 = n 1 180(n – 2) = 108n 180n – 360 = 108n 72n – 360 = 0 72n = 360 n = 5

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