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Math Help - Bisections of parallelogram form a rectangle

  1. #1
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    Bisections of parallelogram form a rectangle

    Hello everyone,

    I am having trouble writing down the exact proofs of these two questions:

    1. The four bisections of a parallelogram form a rectangle.
    2. If the bisections of quadrileteral ABCD form a rectangle, ABCD is a parallelogram.

    First off, here's the analysis figure:
    Bisections of parallelogram form a rectangle-schermafbeelding-2013-03-09-om-17.14.58.png

    For question 1, I have written down the following:

    AB // DC => Angle BAC + Angle ADC = 180o. α + δ = 90o. => Angle ASD = 90o
    The same can now be done with Angle CQB. However, Angles R and P are not in triangles, so I'm stuck there.

    For question 2, I obviously give the reverse proof, starting from Angle ASD being 90o and then reading backwards from question 1, but I have the same problem with angles P and R as I had with question 1.

    If anyone could help me with this, thanks a lot!
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  2. #2
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    Re: Bisections of parallelogram form a rectangle

    \measuredangle BAD=\measuredangle DAS + \measuredangle SAB = 2 \measuredangle DAS

    \measuredangle ADC=\measuredangle ADS + \measuredangle SDC = 2 \measuredangle ADS


    AB || DC \Rightarrow \measuredangle BAD+\measuredangle ADC=180^o \Rightarrow 2 \measuredangle DAS+2 \measuredangle ADS=180^o \Rightarrow

    \Rightarrow  \measuredangle DAS+\measuredangle ADS=90^o


    \measuredangle DAS+\measuredangle ADS+\measuredangle DSA=180^o \Rightarrow \measuredangle DSA =180^o-90^o=90^o\Rightarrow

    \Rightarrow \measuredangle RSP=90^o


    Do you think you can continue from here?
    Last edited by veileen; March 9th 2013 at 01:54 PM.
    Thanks from tomkoolen
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  3. #3
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    Re: Bisections of parallelogram form a rectangle

    Alright thank you very much. When I have proven that angles RSP and BQC are 90 degrees, can I then say that the other 2 angles must be 90 degrees because otherwise a quadrileteral is impossible?
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  4. #4
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    Re: Bisections of parallelogram form a rectangle

    You're welcome!

    Uhm, no. But you can easily prove the other two angles are right angles (triangles DPC and ARB).
    Attached Thumbnails Attached Thumbnails Bisections of parallelogram form a rectangle-aa.png  
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