Question 1041968
I do not see your star, but I think I can make three possible stars by extending the sides of the nonagon.  -0.026,2.82
{{{drawing(645,300,-6.6,2,-2,2,
line(1,0,0.766,0.643),line(0.766,0.643,0.174,0.985),line(0.174,0.985,-0.5,0.866),
line(-0.5,0.866,-0.94,0.342),line(-0.94,0.342,-0.94,-0.342),line(-0.94,-0.342,-0.5,-0.866),
line(-0.5,-0.866,0.174,-0.985),line(0.174,-0.985,0.766,-0.643),line(0.766,-0.643,1,0),
green(arrow(1,0,2.026,-2.819)),green(arrow(0.766,0.643,-0.26,3.462)),
green(arrow(0.766,0.643,3.364,-0.857)),green(arrow(0.174,0.985,-2.424,2.485)),
green(arrow(0.174,0.985,3.128,1.506)),green(arrow(-0.5,0.866,-8.378,-0.523)),
green(arrow(-0.5,0.866,1.428,3.164)),green(arrow(-0.94,0.342,-3.511,-2.722)),
green(arrow(-0.94,0.342,-0.94,3.342)),green(arrow(-0.94,-0.342,-0.94,-3.342)),
green(arrow(-0.94,-0.342,-3.511,2.722)),green(arrow(-0.5,-0.866,1.428,-3.164)),
green(arrow(-0.5,-0.866,-8.378,0.523)),green(arrow(0.174,-0.985,3.128,-1.506)),
green(arrow(0.174,-0.985,-2.424,-2.485)),green(arrow(0.766,-0.643,3.364,0.857)),
green(arrow(0.766,-0.643,-0.26,-3.462)),green(arrow(1,0,2.026,2.819))
)}}}
For an easy problem you would have one of the 3 possible stars: the one that is made by adding isosceles triangles built around the nonagon. Each one of those isosceles triangles has a side of the nonagon as its base.
{{{drawing(645,300,-6.6,2,-2,2,
green(arrow(1,0,2.026,-2.819)),green(arrow(0.766,0.643,-0.26,3.462)),
green(arrow(0.766,0.643,3.364,-0.857)),green(arrow(0.174,0.985,-2.424,2.485)),
green(arrow(0.174,0.985,3.128,1.506)),green(arrow(-0.5,0.866,-8.378,-0.523)),
green(arrow(-0.5,0.866,1.428,3.164)),green(arrow(-0.94,0.342,-3.511,-2.722)),
green(arrow(-0.94,0.342,-0.94,3.342)),green(arrow(-0.94,-0.342,-0.94,-3.342)),
green(arrow(-0.94,-0.342,-3.511,2.722)),green(arrow(-0.5,-0.866,1.428,-3.164)),
green(arrow(-0.5,-0.866,-8.378,0.523)),green(arrow(0.174,-0.985,3.128,-1.506)),
green(arrow(0.174,-0.985,-2.424,-2.485)),green(arrow(0.766,-0.643,3.364,0.857)),
green(arrow(0.766,-0.643,-0.26,-3.462)),green(arrow(1,0,2.026,2.819)),
triangle(1,0,1.153,-0.42,0.766,-0.643),
triangle(0.766,0.643,1.153,0.42,1,0),
triangle(0.174,0.985,0.613,1.062,0.766,0.643),
triangle(-0.5,0.866,-0.213,1.208,0.174,0.985),
triangle(-0.94,0.342,-0.94,0.789,-0.5,0.866),
triangle(-0.94,-0.342,-1.227,0,-0.94,0.342),
triangle(-0.5,-0.866,-0.94,-0.788,-0.94,-0.342),
triangle(0.174,-0.985,-0.213,-1.208,-0.5,-0.866),
triangle(0.766,-0.643,0.613,-1.062,0.174,-0.985)
)}}}
The two base angles of each of those isosceles triangles are exterior angles of the nonagon.
The measure of an exterior angle of a regular polygon with {{{n}}} sides is {{{360^o/n}}} ,
so the measure of an exterior angle of a regular nonagon is {{{360^o/9=40^o}}} .
So, those isosceles triangles have {{{2}}} base angles measuring {{{40^o}}} .
The other angle is at a tip of the start.
Since the sum of the measures of the angles of any triangle is {{{180^o}}} ,
the angle at the tip og the star measures
{{{180^o-2*40^o=180^o-80^o=highlight(100^o)}}} .
 
NOTE:
The other two stars have sharper tips, whose angles are not quite as easy to calculate.
Here is the smaller one:
{{{drawing(645,300,-6.6,2,-2,2,
line(1,0,0.766,0.643),line(0.766,0.643,0.174,0.985),line(0.174,0.985,-0.5,0.866),
line(-0.5,0.866,-0.94,0.342),line(-0.94,0.342,-0.94,-0.342),line(-0.94,-0.342,-0.5,-0.866),
line(-0.5,-0.866,0.174,-0.985),line(0.174,-0.985,0.766,-0.643),line(0.766,-0.643,1,0),
green(arrow(1,0,2.026,-2.819)),green(arrow(0.766,0.643,-0.26,3.462)),
green(arrow(0.766,0.643,3.364,-0.857)),green(arrow(0.174,0.985,-2.424,2.485)),
green(arrow(0.174,0.985,3.128,1.506)),green(arrow(-0.5,0.866,-8.378,-0.523)),
green(arrow(-0.5,0.866,1.428,3.164)),green(arrow(-0.94,0.342,-3.511,-2.722)),
green(arrow(-0.94,0.342,-0.94,3.342)),green(arrow(-0.94,-0.342,-0.94,-3.342)),
green(arrow(-0.94,-0.342,-3.511,2.722)),green(arrow(-0.5,-0.866,1.428,-3.164)),
green(arrow(-0.5,-0.866,-8.378,0.523)),green(arrow(0.174,-0.985,3.128,-1.506)),
green(arrow(0.174,-0.985,-2.424,-2.485)),green(arrow(0.766,-0.643,3.364,0.857)),
green(arrow(0.766,-0.643,-0.26,-3.462)),green(arrow(1,0,2.026,2.819)),
red(line(0.612,1.066,0.326,1.851)),red(line(-0.211,1.211,0.326,1.851)),
red(line(-0.216,1.21,-0.94,1.628)),red(line(-0.94,0.792,-0.94,1.628)),
red(line(-1.229,0.003,-1.766,0.643)),red(line(-0.943,0.788,-1.766,0.643)),
red(line(-0.943,-0.788,-1.766,-0.643)),red(line(-1.229,-0.003,-1.766,-0.643)),
red(line(-0.94,-0.792,-0.94,-1.628)),red(line(-0.216,-1.21,-0.94,-1.628)),
red(line(0.612,-1.066,0.326,-1.851)),red(line(-0.211,-1.211,0.326,-1.851)),
red(line(1.154,-0.423,1.44,-1.208)),red(line(0.617,-1.063,1.44,-1.208)),
red(line(1.156,0.418,1.88,0)),red(line(1.156,-0.418,1.88,0)),
red(line(0.617,1.063,1.44,1.208)),red(line(1.154,0.423,1.44,1.208))
)}}} .
The larger one, with the sharpest tips is too big A partial drawing is shown below.
{{{drawing(645,300,-6.6,2,-2,2,
line(1,0,0.766,0.643),line(0.766,0.643,0.174,0.985),line(0.174,0.985,-0.5,0.866),
line(-0.5,0.866,-0.94,0.342),line(-0.94,0.342,-0.94,-0.342),line(-0.94,-0.342,-0.5,-0.866),
line(-0.5,-0.866,0.174,-0.985),line(0.174,-0.985,0.766,-0.643),line(0.766,-0.643,1,0),
green(arrow(1,0,2.026,-2.819)),green(arrow(0.766,0.643,-0.26,3.462)),
green(arrow(0.766,0.643,3.364,-0.857)),green(arrow(0.174,0.985,-2.424,2.485)),
green(arrow(0.174,0.985,3.128,1.506)),green(arrow(-0.5,0.866,-8.378,-0.523)),
green(arrow(-0.5,0.866,1.428,3.164)),green(arrow(-0.94,0.342,-3.511,-2.722)),
green(arrow(-0.94,0.342,-0.94,3.342)),green(arrow(-0.94,-0.342,-0.94,-3.342)),
green(arrow(-0.94,-0.342,-3.511,2.722)),green(arrow(-0.5,-0.866,1.428,-3.164)),
green(arrow(-0.5,-0.866,-8.378,0.523)),green(arrow(0.174,-0.985,3.128,-1.506)),
green(arrow(0.174,-0.985,-2.424,-2.485)),green(arrow(0.766,-0.643,3.364,0.857)),
green(arrow(0.766,-0.643,-0.26,-3.462)),green(arrow(1,0,2.026,2.819)),


blue(line(1.44,-1.208,2.706,-4.687)),blue(line(0.327,-1.851,2.706,-4.687)),
blue(line(1.88,0,5.086,-1.851)),blue(line(1.44,-1.208,5.086,-1.8518)),
blue(line(1.44,1.208,5.086,1.851)),blue(line(1.88,0,5.086,1.851)),
blue(line(0.327,1.851,2.706,4.687)),blue(line(1.44,1.208,2.706,4.687)),
blue(line(-0.94,1.628,-0.94,5.33)),blue(line(0.326,1.851,-0.94,5.33)),
blue(line(-1.766,0.643,-4.146,3.479)),blue(line(-0.94,1.628,-4.146,3.479)),
blue(line(-1.766,-0.643,-5.412,0)),blue(line(-1.766,0.643,-5.412,0)),
blue(line(-0.94,-1.628,-4.146,-3.479)),blue(line(-1.766,-0.643,-4.146,-3.479)),
blue(line(0.326,-1.851,-0.94,-5.33)),blue(line(-.94,-1.628,-0.94,-5.33))
)}}}