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Thread: Modern Geometry: Complete SsA proof.

  1. #1
    Member ilikedmath's Avatar
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    Question Modern Geometry: Complete SsA proof.

    Below is the original problem. After doing some online research, I came to the conclusion that an actual "SSA" Congruence does not exist, so I was a little confused with what this problem was asking. Anyway, below the picture are my attempts at the answers for the missing parts:



    My work so far:
    (a) I don't know how these two angles can be supplementary . I did find a corollary that states, "The sum of the measrues of 2 angels of a triangle is less than 180."

    (b) XY > YZ

    (c) Contradiction to the given that BC ≥ BA.

    * I think I got (b) and (c), but part (a) is still throwing me off.

    Thank you for your time, help, suggestions/corrections.
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  2. #2
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    Opalg's Avatar
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    This depends crucially on the condition $\displaystyle BC\geqslant BA$. Notice that if the length BC is less than BA then the triangles XYZ and XYZ' in this picture would both have the same SSA data as ABC (where YZ'=YZ=BC), although XYZ' is not congruent to ABC:

    $\displaystyle \setlength{\unitlength}{6mm}
    \begin{picture}(10,10)
    \put(0.5,2){\line(1,0){9}}
    \put(0.5,2){\line(3,2){6}}
    \put(9.5,2){\line(-3,4){3}}
    \put(3.5,2){\line(3,4){3}}
    \put(0.4,1.25){$X$}
    \put(3.4,1.25){$Z'$}
    \put(9.4,1.25){$Z$}
    \put(6.4,6.25){$Y$}
    \end{picture}$

    That is a typical SSA situation, where the position of Z is ambiguous.

    But if $\displaystyle BC\geqslant BA$ then the position of Z' would be to the left of X and the triangle XYZ would look like this:

    $\displaystyle \setlength{\unitlength}{6mm}
    \begin{picture}(15,10)
    \put(0.5,2){\line(1,0){12}}
    \put(0.5,2){\line(6,5){6}}
    \put(12.5,2){\line(-6,5){6}}
    \put(4.5,2){\line(2,5){2}}
    \put(0,1.25){$Z'$}
    \put(4,1.25){$X$}
    \put(12,1.25){$Z$}
    \put(6.25,7.25){$Y$}
    \end{picture}$

    Notice that in the triangle XYZ' the angle at X is now the supplement of what it is in the triangle XYZ. So the triangle XYZ' does not have the same SSA data as ABC in this case.

    I think that is why the question asks for an "SsA" proof, with the second s in lower case. It's essential that the side AB should be shorter than BC.
    Last edited by Opalg; Mar 21st 2009 at 01:13 AM. Reason: corrected error
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