Question 233700
<pre><font size = 4 color = "indigo"><b>
All vertex angles (also called "interior angles") of a regular
polygon have equal measures. If this regular polygon has n sides,
then the sum of all the interior angles is 150° times n. The sum
of the measures of all the interior angles of any polygon with n
sides is (n-2) times 180°. So we have this equation

{{{ 150n=(n-2)(180) }}}

Solve that and we get {{{n=12}}}

So we have a 12-sided regular polygon (a regular dodecagon)

{{{ drawing(400,400,-1.5,1.5,-1.5,1.5,
line(1,0,.8660254038,.5),line(.8660254038,.5,.5,.8660254038)  ,
line(.5,.8660254038,0,1)    ,
line(0,1,-.5,.8660254038)    ,
line(-.5,.8660254038,-.8660254038,.5)   ,
line(-.8660254038,.5,-1,0)      ,
line(-1,0,-.8660254038,-.5)     ,
line(-.8660254038,-.5,-.5,-.8660254038)   ,
line(-.5,-.8660254038,0,-1)   ,
line(0,-1,.5,-.8660254038),
line(.5,-.8660254038,.8660254038,-.5),
line(.8660254038,-.5,1,0) )}}}

Now we'll connect all the vertices to the center,

{{{ drawing(400,400,-1.5,1.5,-1.5,1.5,
line(1,0,.8660254038,.5),line(.8660254038,.5,.5,.8660254038)  ,
line(.5,.8660254038,0,1)    ,
line(0,1,-.5,.8660254038)    ,
line(-.5,.8660254038,-.8660254038,.5)   ,
line(-.8660254038,.5,-1,0)      ,
line(-1,0,-.8660254038,-.5)     ,
line(-.8660254038,-.5,-.5,-.8660254038)   ,
line(-.5,-.8660254038,0,-1)   ,
line(0,-1,.5,-.8660254038),
line(.5,-.8660254038,.8660254038,-.5),
line(.8660254038,-.5,1,0),

line(1,0,0,0)      ,
line(.8660254038,.5,0,0)  ,
line(.5,.8660254038,0,0)    ,
line(0,1,0,0)    ,
line(-.5,.8660254038,0,0)   ,
line(-.8660254038,.5,0,0)      ,
line(-1,0,0,0)     ,
line(-.8660254038,-.5,0,0)   ,
line(-.5,-.8660254038,0,0)   ,
line(0,-1,0,0),
line(.5,-.8660254038,0,0),
line(.8660254038,-.5,0,0)

 )}}}

There are 12 central angles all
with equal measure and since they
all add up to 360°, each one is {{{1/12}}}
of 360° or 30°.

Edwin</pre>