Question 979233
The following polygons are given. All of the polygons are 
regular polygons.

Polygon a. Convex 15-gon
Polygon b. Convex 16-gon
Polygon c. Convex 17-gon
Polygon d. Convex 18-gon
Polygon e. Convex 19-gon
Polygon f. Convex 43-gon
Polygon g. Convex 44-gon
Polygon h. Convex 45-gon
Polygon i. Convex 46-gon
Polygon j. Convex 47-gon

1. Which polygon(s) has (have) interior angles that are whole 
numbers (a number that is not a fraction or a decimal)? Explain 
why it is that way.
<pre>
The sum of the interior angles of a polygon of n-sides is

{{{(n-2)*"180°"}}}

Since the polygons are regular, all the interior angles are the same,
so each one is that expression divided by n

{{{(n-2)*"180°"/n}}}

That must be equal to a whole number, say, W. Since n does not divide
evenly into n-2, it must divide evenly into 180°. So we go through
the list to see which numbers divide evenly into 180°:

Polygon a. Convex 15-gon, yes, since 15 divides evenly into 180°.
Polygon b. Convex 16-gon, no
Polygon c. Convex 17-gon, no
Polygon d. Convex 18-gon, yes, since 18 divides evenly into 180°.
Polygon e. Convex 19-gon, no
Polygon f. Convex 43-gon, no
Polygon g. Convex 44-gon, no
Polygon h. Convex 45-gon, yes, since 45 divides evenly into 180°.
Polygon i. Convex 46-gon, no
Polygon j. Convex 47-gon, no
</pre>
2. What happens to the value of the interior angles as the 
number of sides of the polygon increases? Explain your answer.
<pre>
{{{matrix(2,2,

lim,((n-2)*"180°")/n,
n-"">infinity,"")}}}{{{""=""}}}{{{"180°"*matrix(2,2,

lim,(n-2)/n,
n-"">infinity,"")}}}{{{""=""}}}{{{"180°"*1}}}{{{""=""}}}{{{"180°"}}}

So the value of the interior angles approaches 180° as the number 
of sides of the polygon increases.
</pre>
3. What happens to the value of the exterior angles as the 
number of sides of the polygon increases? Explain your answer.
<pre>
The sum of the exterior angles of any polygon is 360°.  So
each one of a regular polygon is {{{("360°")/n

{{{matrix(2,2,

lim,("360°")/n,
n-"">infinity,"")}}}{{{""=""}}}{{{"0°"}}}

So the value of the exterior angles approaches 0° as the number 
of sides of the polygon increases.
</pre>
4. Explain what happens to the total sum of interior angles 
as the number of sides in the polygon changes?
<pre>{{{matrix(2,2,
lim,((n-2)*"180°"),
n-"">infinity,"")}}}{{{""=""}}}{{{infinity}}}
</pre>
5. Explain what happens to the total sum of exterior angles 
as the number of sides in the polygon changes?
<pre>{{{matrix(2,2,
lim,("360°"),
n-"">infinity,"")}}}{{{""=""}}}{{{"360°"}}}

Edwin</pre>