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anonymous

  • one year ago

#17 If the vertex angle of a regular polygon has measure 168 degrees, how many sides does it have?

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  1. anonymous
    • one year ago
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  2. mathstudent55
    • one year ago
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    There is a formula that relates the number of sides of a polygon to the sum of the measures of the interior angles.

  3. anonymous
    • one year ago
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    Is that (n-2)180

  4. anonymous
    • one year ago
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    ?

  5. mathstudent55
    • one year ago
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    \(S = (n - 2)180\) where S = sum of the measures of the interior angles n = number of isdes of the polygon

  6. mathstudent55
    • one year ago
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    Exactly.

  7. anonymous
    • one year ago
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    let me try to solve real quick.

  8. mathstudent55
    • one year ago
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    What do you know about the sides and angles of a regular polygon?

  9. anonymous
    • one year ago
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    they're congruent right?

  10. mathstudent55
    • one year ago
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    Yes. All angles are congruent.

  11. mathstudent55
    • one year ago
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    If all angles are congruent, and you know the sum of the measures of the angles, if you divide the sum by the number of angles, you get the measure of each angle, right?

  12. anonymous
    • one year ago
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    right

  13. mathstudent55
    • one year ago
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    Here is an example. Think of a square. A square is a regular polygon with 4 sides. The sum of the measures of the interior angles of a square is: \(S = (n - 2)180 = (4 - 2)180 = 2(180) = 360\) Now we divide 360 by the number of angles, 360/4 = 90 That means each angle of a square measures 90 degrees. That makes sense because we know a square has 4 right angles.

  14. anonymous
    • one year ago
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    right

  15. mathstudent55
    • one year ago
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    Now think of your problem. The unknown in your problem is the number of sides, n. The sum of the measures of the angles is \((n - 2)180\) If we divide that sum by the number of angles, we get the measure of each angle: \(each~angle~measure = \dfrac{(n - 2)180}{n} \)

  16. mathstudent55
    • one year ago
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    \( \dfrac{(n - 2)180}{n} = 168\)

  17. mathstudent55
    • one year ago
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    We first multiply both sides by n. Now think of your problem. The unknown in your problem is the number of sides, n. The sum of the measures of the angles is \((n - 2)180\) If we divide that sum by the number of angles, we get the measure of each angle: \((n - 2)180 = 168n \)

  18. mathstudent55
    • one year ago
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    We rewrite the left side: \(180(n - 2) = 168n\) We distribute the 180 on the left side: \(180n - 360 = 168n\)

  19. anonymous
    • one year ago
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    360/180?

  20. anonymous
    • one year ago
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    wait nvm

  21. mathstudent55
    • one year ago
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    Now we subtract 168n from both sides: \(12n - 360 = 0\) We add 360 to both sides: \(12n = 360\) Divide both sides by 12: \(n = 30\) The polygon has 30 sides.

  22. anonymous
    • one year ago
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    Much appreciated.

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