SOLUTION: Please help me answer or solve this question: How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle? It wo

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Question 109127: Please help me answer or solve this question:
How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle?
It would be very helpful if you could explain the process that you took to solve this problem.
Thanks in advance,
Christine
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle?
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Draw the figure of an exterior angle and its corresponding
interior angle. Notice that they are supplementary.
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Let the exterior angle have measure "x"; The interior angle has measure "8x".
EQUATION:
x + 8x = 180
9x = 180
x = 20 degrees (measure of one of the exterior angles.
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Fact: The sume of the exterior angles of a polygon is 360 degrees.
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EQUATION:
# of sides = 360/20 = 18 sides
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Cheers,
Stan H.

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

Please help me answer or solve this question:
How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle?
It would be very helpful if you could explain the process that you took to solve this problem.
Thanks in advance,
Christine 

The drawing below shows the bottom right-hand corner of the polygon:

  

The small acute angle marked x is an exterior angle of the
polygon and the large obtuse angle marked 8x is an interior 
angle. It measures 8x because we are told that each interior 
angle is 8 times the measure of each exterior angle. 

Since the interior and exterior angle at the same vertex
of the polygon are supplementary, we can equate their sum to
180°.

INTERIOR ANGLE + EXTERIOR ANGLE = 180°

                         8x + x = 180°
                             9x = 180°
                              x = 20°

This means each exterior angle is 20° and each interior
angle is 180°-20° or 160°. This tells us that the polygon
is a regular polygon because all the interior angles are
equal in measure. 

Now we only need to know that the theorem that says: 

The sum of all the exterior angles of any polynomial is 
always 360°.  

Let the number of sides of this polygon be N.

Since we know that this is an N-sided regular polygon, 
the sum of the exterior angles is N times 20° or 20N
degrees. So 

            20N degrees = 360 degrees
                    20N = 360
                      N = 18 sides

So it is an 18-sided regular polygon. 
                    
Edwin