Bunuel wrote:
What is the value of x in the figure above?
(1) x > 40
(2) x = y
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\(? = x\)
Let´s start with (2), because (we believe) it´s (at least at first sight) much easier!
\(\left( 2 \right)\,\,\left\{ \begin{gathered}
\,x = y \hfill \\
\,x + y + 40 = 180 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,2x + 40 = 180\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,x\,\,{\text{unique}}\,\,\,\)
\(\left( 1 \right)\,\,\,{\text{Geometric}}\,\,{\text{Bifurcation}}\, (below)\,\)
Obs.: the figure on the left is "visually compatible" to statement (2), in which x is obviously 70 (2x+40=180), therefore the figure on the left (the one presented in the question stem) is viable (70>40). The figure on the right makes x greater, hence also > 40.
Both figures are viable, that is, they are constructible (with straight-edge and compass) AND they satisfy both the question stem pre-statements and the statement considered... the geometric bifurcation is shielded!
(We follow the notations and rationale taught in the GMATH method.)
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