Question 1150799
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From one side, the sum of interior angles of any 9-gon is  (9-2)*180 = 7*180 degrees.


From the other side, the sum of the first n terms of any arithmetic progression is


    {{{S[n]}}} = {{{((a[1]+a[n])/2)*n}}}.


In our case,  {{{((a[1]+a[9])/2)*9}}} = 7*180,  so


    {{{(a[1]+a[9])/2}}} = 140  degrees.


For arithmetic progression, the average of any two terms, equally remoted from the central term, is the same.


In particular, the central term  {{{a[5]}}} is 140 degrees.


The terms of the AP are


    {{{a[4]}}} = 140 -  d,   {{{a[6]}}} = 140 +  d,

    {{{a[3]}}} = 140 - 2d,   {{{a[7]}}} = 140 + 2d,

    {{{a[2]}}} = 140 - 3d,   {{{a[8]}}} = 140 + 3d,

    {{{a[1]}}} = 140 - 4d,   {{{a[9]}}} = 140 + 4d.


To make {{{a[1]}}} as small as possible, we should take the common difference as large as possible.


We have two constraints:  {{{a[1]}}} must be positive,  {{{a[1]}}} > 0,                                      (1)

and
                          {{{a[9]}}} must be less than 180 degrees;  so  {{{a[9]}}}  must be 176 degrees.    (2)


Of these two constraints, the constraint (2) is more cumbersome, and it gives

    d = 36/4 = 9 degrees.


Then both constraints (1) and (2) are satisfied.


Thus the minimum angle is  {{{a[1]}}} = 140 - 4*9 = 140 - 36 = 104 degrees.    <U>ANSWER</U>
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