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