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

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

Mathematics
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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.
Is that (n-2)180

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

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