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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**File comment:** Geometric Bifurcation - GMATH method

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Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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