anonymous
  • anonymous
#17 If the vertex angle of a regular polygon has measure 168 degrees, how many sides does it have?
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
mathstudent55
  • mathstudent55
There is a formula that relates the number of sides of a polygon to the sum of the measures of the interior angles.
anonymous
  • anonymous
Is that (n-2)180

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anonymous
  • anonymous
?
mathstudent55
  • mathstudent55
\(S = (n - 2)180\) where S = sum of the measures of the interior angles n = number of isdes of the polygon
mathstudent55
  • mathstudent55
Exactly.
anonymous
  • anonymous
let me try to solve real quick.
mathstudent55
  • mathstudent55
What do you know about the sides and angles of a regular polygon?
anonymous
  • anonymous
they're congruent right?
mathstudent55
  • mathstudent55
Yes. All angles are congruent.
mathstudent55
  • mathstudent55
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?
anonymous
  • anonymous
right
mathstudent55
  • mathstudent55
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.
anonymous
  • anonymous
right
mathstudent55
  • mathstudent55
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: \(each~angle~measure = \dfrac{(n - 2)180}{n} \)
mathstudent55
  • mathstudent55
\( \dfrac{(n - 2)180}{n} = 168\)
mathstudent55
  • mathstudent55
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: \((n - 2)180 = 168n \)
mathstudent55
  • mathstudent55
We rewrite the left side: \(180(n - 2) = 168n\) We distribute the 180 on the left side: \(180n - 360 = 168n\)
anonymous
  • anonymous
360/180?
anonymous
  • anonymous
wait nvm
mathstudent55
  • mathstudent55
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.
anonymous
  • anonymous
Much appreciated.

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