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Math Help - 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
    MHF Contributor
    Opalg's Avatar
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    This depends crucially on the condition 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:

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

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

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

    \setlength{\unitlength}{6mm}<br />
\begin{picture}(15,10)<br />
\put(0.5,2){\line(1,0){12}}<br />
\put(0.5,2){\line(6,5){6}}<br />
\put(12.5,2){\line(-6,5){6}}<br />
\put(4.5,2){\line(2,5){2}}<br />
\put(0,1.25){$Z'$}<br />
\put(4,1.25){$X$}<br />
\put(12,1.25){$Z$}<br />
\put(6.25,7.25){$Y$}<br />
\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; March 21st 2009 at 02:13 AM. Reason: corrected error
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