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It's all in the stars (Posted on 2016-12-14) Difficulty: 2 of 5
It’s easy to show that the five acute angles in the vertices
of a regular 5-pointed star total 180°.

Please show that the sum of these angles in an irregular 5-pointed star is also 180°.

Source: A. Korshkov, the Russian science magazine Kvant.

No Solution Yet Submitted by Ady TZIDON    
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Solution Solution | Comment 1 of 2


  Let A, B, C, D, and E label (in that order) the five points
  of the star.

  Let I = AC /\ BE, 
         J = BD /\ CA,
         K = CE /\ DB,      /\  denotes intersection
         L = DA /\ EC,
         M = EB /\ AD.

   5*180 is the sum (in degrees) of the interior angles of the 
               following five triangles: CIE, DJA, EKB, ALC, and 
              BMD.

   3*180 is the sum (in degrees) of the interior angles of the
               pentagon IJKLM.

   Star is the sum (in degrees) of angles A, B, C, D, and E.

   Therefore,

       2*Star + 3*180 = 5*180

                       or

       Star = 180 degrees.

   QED 



  Posted by Bractals on 2016-12-14 13:15:45
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