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Math Help - 5 pointed star

  1. #1
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    Exclamation 5 pointed star

    what observations can you make about the sum of the angles of five pointed stars - How can you prove your observation?
    I know the sum of 5 pointed stars is 180 degrees. Other than saying you just add up the interior angles, is there another way I can offer "proof"?? THANK YOU!!
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  2. #2
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    sketch the circle that contains all 5 "points".

    now use the fact that an angle inscribed in a circle equals one-half the measure of its intercepted arc.
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  3. #3
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    THANKS!!!
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  4. #4
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    Hello, jmccoid!

    What observations can you make about the sum of the angles of five pointed stars?
    How can you prove your observation?
    Is this a cyclic pentagram? (not necessarily regular)
    If so, it can be inscribed in a circle and skeeter's solution is excellent.
    Code:
                    A
                  * o *
              *           *
            *               *
         E o                 o B
    
          *                   *
          *                   *
          *                   *
    
           *                 *
          D o               o C
              *           *
                  * * *

    Draw the diagonals: . AC,\, AD,\, BD,\, BE,\, CE

    An incribed angle is measured by one-half its intercepted arc.

    . . \begin{array}{ccc}\angle A &^m_= &\tfrac{1}{2}\overline{CD}  \\ \\[-3mm]<br />
\angle B & ^m_= & \tfrac{1}{2}\overline{DE} \\ \\[-3mm]<br />
\angle C & ^m_= & \tfrac{1}{2}\overline{EA} \\ \\[-3mm]<br />
\angle D & ^m_= & \tfrac{1}{2}\overline{AB} \\ \\[-3mm]<br />
\angle E & ^m_= & \tfrac{1}{2}\overline{BC}<br />
\end{array}


    Add: . \angle A + \angle B + \angle C + \angle D + \angle E \;=\;\tfrac{1}{2}\overline{CD} + \tfrac{1}{2}\overline{DE} + \tfrac{1}{2}\overline{EA} + \tfrac{1}{2}\overline{AB} + \tfrac{1}{2}\overline{BC}


    \text{Sum of the angles} \;=\;\tfrac{1}{2}\left(\overline{AB} + \overline{BC} + \overline{CD} + \overline{DE} + \overline{EA}\right) \;=\;\tfrac{1}{2}(360^o) \;=\;180^o

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