SOLUTION: How can you find the sum of the interior angles of the points of a 5 point star made out of three triangles, but forming a polygon? There are no equilateral triangles and no right

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Question 225075: How can you find the sum of the interior angles of the points of a 5 point star made out of three triangles, but forming a polygon? There are no equilateral triangles and no right angles.
Answer by Edwin McCravy(6934) About Me  (Show Source):
You can put this solution on YOUR website!
How can you find the sum of the interior angles of the points of a 5 point star made out of three triangles, but forming a polygon? There are no equilateral triangles and no right angles.

I'm not sure what you mean.  Here's a 5-pointed star, but I don't
know how to make a 5-pointed star out of three triangles.

Did you mean this kind of 5-pointed star, also known as a pentagram?  

drawing%28350%2C350%2C-3%2C3%2C-3%2C3%2C%0D%0A%0D%0Aline%280%2C2.618033989%2C.5877852523%2C.8090169944%29%2C%0D%0Aline%28.5877852523%2C.8090169944%2C2.4898983%2C.80901699%29%2C%0D%0Aline%282.4898983%2C.80901699%2C.9510565163%2C-.3090169944%29%2C%0D%0Aline%28.9510565163%2C-.3090169944%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%281.5388468%2C-2.118034%2C0%2C-1%29%2C%0D%0A%0D%0Aline%280%2C2.618033989%2C-.5877852523%2C.8090169944%29%2C%0D%0Aline%28-.5877852523%2C.8090169944%2C-2.4898983%2C.80901699%29%2C%0D%0Aline%28-2.4898983%2C.80901699%2C-.9510565163%2C-.3090169944%29%2C%0D%0Aline%28-.9510565163%2C-.3090169944%2C-1.5388468%2C-2.118034%29%2C%0D%0Aline%28-1.5388468%2C-2.118034%2C0%2C-1%29+%29

If so, we'll extend the lines like this:

drawing%28350%2C350%2C-3%2C3%2C-3%2C3%2C%0D%0Aline%280%2C2.618033989%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%280%2C2.618033989%2C-1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-2.4898983%2C.80901699%29%2C%0D%0Aline%28-2.4898983%2C.80901699%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-1.5388468%2C-2.118034%29+%29

In the center is a 5-sided regular polygon (a regular
pentagon).  The sum of the interior angles of a polygon
is gotten by the formula:

SUM OF INTERIOR ANGLES = (NUMBER OF SIDES - 2) * 180°

For a 5-sided polygon (pentagon) this is (5-2)*180° = 3*180°=540°

Since all 5 angles of a regular pentagon are equal, each
interior angle of the regular pentagon is 540%2F5° = 108°

So I'll mark one of the 108° interior angles of the pentagon:

drawing%28350%2C350%2C-3%2C3%2C-3%2C3%2C%0D%0Aline%280%2C2.618033989%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%280%2C2.618033989%2C-1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-2.4898983%2C.80901699%29%2C%0D%0Aline%28-2.4898983%2C.80901699%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-1.5388468%2C-2.118034%29%2C%0D%0Alocate%28-.6%2C.7%2C%27108%B0%27%29%0D%0A+%29

Its suppplement is found by subtracting 180°-108°=72°.
We'll mark it 72°:

drawing%28350%2C350%2C-3%2C3%2C-3%2C3%2C%0D%0Aline%280%2C2.618033989%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%280%2C2.618033989%2C-1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-2.4898983%2C.80901699%29%2C%0D%0Aline%28-2.4898983%2C.80901699%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-1.5388468%2C-2.118034%29%2C%0D%0Alocate%28-.6%2C.7%2C%27108%B0%27%29%2C+locate%28-.45%2C1.1%2C%2772%B0%27%29%0D%0A+%29

That 72° angle is one of the base angles of an isosceles
triangle.  So we'll mark the other base angle 72° also.

drawing%28350%2C350%2C-3%2C3%2C-3%2C3%2C%0D%0Aline%280%2C2.618033989%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%280%2C2.618033989%2C-1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-2.4898983%2C.80901699%29%2C%0D%0Aline%28-2.4898983%2C.80901699%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-1.5388468%2C-2.118034%29%2C%0D%0Alocate%28-.6%2C.7%2C%27108%B0%27%29%2Clocate%28-.45%2C1.1%2C%2772%B0%27%29%0D%0Alocate%280%2C1.1%2C%2772%B0%27%29+%29

Now we can find the angle at the top point of the star by
adding the two equal base angles and subtracting from 180°.

72° + 72° = 144°
180° - 144° = 36°

So each point of the star is 36°.

drawing%28350%2C350%2C-3%2C3%2C-3%2C3%2C%0D%0Aline%280%2C2.618033989%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%280%2C2.618033989%2C-1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-2.4898983%2C.80901699%29%2C%0D%0Aline%28-2.4898983%2C.80901699%2C1.5388468%2C-2.118034%29%2C%0D%0Aline%282.4898983%2C.80901699%2C-1.5388468%2C-2.118034%29%2C%0D%0Alocate%28-.6%2C.7%2C%27108%B0%27%29%2Clocate%28-.45%2C1.1%2C%2772%B0%27%29%0D%0Alocate%280%2C1.1%2C%2772%B0%27%29%2C+locate%28-.16%2C2%2C%2736%B0%27%29++%29


You wanted the sum of the points interior angles of 
the points.  There are 5 of them, so 5 times 36° is

180°.

Edwin