Question 940570
The exterior angle and the interior angle are supplementary,
meaning that they add up to {{{180^o}}} .
Let {{{x}}} be the measure of the exterior angle in degrees.
Then, {{{180-x}}} is the measure of the interior angle in degrees, and
"each interior angle is exactly 9 times as large as the matching exterior angle" translates as
{{{180-x=9x}}}<--->{{{180=9x+x}}}<--->{{{180=10x}}}<--->{{{180/10=x}}}<--->{{{x=18}}} .
The exterior angle is the angle you turn around the corner/vertex, as you go around the polygon. Naturally, once you went all the way around, you have turned {{{360^o}}} , so the sum of the measures of all exterior angles is {{{360^o}}} .
All the exterior angles in a regular polygon have the same measure,
so if each exterior angle measures {{{18^o}}} ,
the sum of the exterior angles in the polygon is
{{{(180^o)*n=360^o}}}<--->{{{n=360^o/180^o}}}<--->{{{n=20}}} .
It is possible to make a polygon with 20 sides angles, so the answer is {{{highlight(YES)}}} .
If we had found a result for {{{n}}} that was not a whole number, or was a whole number less than 3, the answer would have been no.