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

Question

anonymous · Answer

attachment

mathstudent55 · Answer

There is a formula that relates the number of sides of a polygon to the sum of the measures of the interior angles.

anonymous · Answer

Is that (n-2)180

anonymous · Answer

?

mathstudent55 · Answer

$S = (n - 2)180$

where S = sum of the measures of the interior angles
n = number of isdes of the polygon

mathstudent55 · Answer

Exactly.

anonymous · Answer

let me try to solve real quick.

mathstudent55 · Answer

What do you know about the sides and angles of a regular polygon?

anonymous · Answer

they're congruent right?

mathstudent55 · Answer

Yes.
All angles are congruent.

mathstudent55 · Answer

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 · Answer

right

mathstudent55 · Answer

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 · Answer

right

mathstudent55 · Answer

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 · Answer

$ \dfrac{(n - 2)180}{n} = 168$

mathstudent55 · Answer

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 · Answer

We rewrite the left side:

$180(n - 2) = 168n$

We distribute the 180 on the left side:

$180n - 360 = 168n$

anonymous · Answer

360/180?

anonymous · Answer

wait nvm

mathstudent55 · Answer

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 · Answer

Much appreciated.