anonymous
  • anonymous
Find the number of sides of a regular polygon if one interior angle is 120 degrees. A.) 6 B.) 5 C.) 4 D.) 3
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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mathstudent55
  • mathstudent55
Do you know the formula that tells you the sum of the measures of the interior angles of a polygon?
anonymous
  • anonymous
No
mathstudent55
  • mathstudent55
Here is the formula: \(S = (n - 2)180\) where S = sum of the measures of the interior angles, and n = number of sides.

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mathstudent55
  • mathstudent55
Do you know what "regular" means when you say regular polygon?
anonymous
  • anonymous
When all angles are equal?
mathstudent55
  • mathstudent55
It means a polygon that has all sides of the same length and all interior angles of the same measure.
mathstudent55
  • mathstudent55
Correct. All angles are congruent, and all sides are congruent.
anonymous
  • anonymous
So would it be 3?
mathstudent55
  • mathstudent55
Now we can solve this problem. Since you are dealing with a regular polygon, all angles have the same measure. Let's say the sum of the measures of the interior angles is S. Also, let's say your polygon has n sides. Then each angle would have the measure S/n Ok so far?
mathstudent55
  • mathstudent55
No, it's not 3.
mathstudent55
  • mathstudent55
Just follow what I am explaining . We'll find the answer soon.
anonymous
  • anonymous
Okay!
mathstudent55
  • mathstudent55
If the sum of the measures of the interior angles is S and the polygon has n sides, then each interior angle measures S/n.
mathstudent55
  • mathstudent55
We have a formula for the sum of the measures of the interior angles. \(S = (n - 2)180\) If we divide it by n, the number of sides, we will have the measure of one angle. \(\dfrac{S}{n} = \dfrac{(n - 2)180}{n} \) is the measure of one angle. We are told one angle measures 120 deg, so we set our expression equal to 120 and ewe solve for n. \(\dfrac{(n - 2)180}{n} = 120\)
anonymous
  • anonymous
I think I got it now! thank you so much!
mathstudent55
  • mathstudent55
Now we have to solve that equation for n. We multiply both sides by n: \((n - 2)180 = 120n\) Distribute 180 on the left side: \(180n - 360 = 120n\) Subtract 120n from both sides, and add 360 to both sides: \(60n = 360\) Divide both sides by 60: \(n = 6\) Since n is the number of sides of the polygon, we can answer: The polygon has 6 sides.
mathstudent55
  • mathstudent55
You're welcome.

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