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Math Help - Nonagon Problem

  1. #1
    Junior Member PythagorasNeophyte's Avatar
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    Nonagon Problem

    I have a question about polygons. Here's how it goes:

    Attachment 19017

    I know that the sum of interior angles of the regular nonagon is 180(n-2)=
    180n -360
    = 180(9) -360
    =1260

    So each of the interior angle is 1260/9 = 140

    angle PCD + angle BCD = 180 (adjacent angles on a straight line)

    angle PCD = 180 - 140

    angle PCD = 40

    So that explains part a).

    Can you please give me tips on part b)?
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  2. #2
    Member Traveller's Avatar
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    Here are some hints :

    i) Triangle DCE is isosceles.

    ii) You will get this from the previous part.

    iii) Triangle BEP is isosceles.
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  3. #3
    Junior Member PythagorasNeophyte's Avatar
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    Hello Traveller.

    Thank you for your reply, but what are the proofs that triangle DCE and trianglle BEP are isosceles triangles?
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  4. #4
    Senior Member Educated's Avatar
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    This is a regular nonagon, which means that all side lengths are the same.

    Triangle DCE uses 2 lengths of the regular nonagon, therefore it must be an iscoceles triangle.

    Triangle BEP are isoceles because length BP and EP both use extended lines of the regular nonagon sides. These lines that meet have to be equal, because the angles of the exterior are the same.

    And also all regular polygons have symmetry, so if you drew a line from one corner to the other, the angles would be the same, thus making it an iscoceles triangle.
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  5. #5
    Junior Member PythagorasNeophyte's Avatar
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    Thank you so much Educated!

    So the answer for b)i) will be:

    CD = DE (sides of isos. triangle)

    Angle CDE is 140 (one of the int. angles of nonagon)

    Angle CDE + angle DCE + angle DEC = 180 (angle sum of triangle)

    Angle DCE + angle DEC = 40

    Angle DCE = 20

    ----------
    As for part ii):

    Angle BCD is 140 .
    Angle DCE is 20 .

    Angle BCE is 140 -20 =120

    ------------
    As for part iii):

    Angle BEF + Angle DEB = angle DEF

    So
    Angle BEF = 140 -2(20 )<br />
              =100

    Problem solved.
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