BTGmoderatorDC wrote:Is quadrilateral ABCD a square?
(1) A = B = C = 90º
(2) AB = AD
Source: Magoosh
\[ABCD\,\,\mathop = \limits^? \,\,{\text{square}}\]
\[\left( 1 \right)\,\,\mathop \Rightarrow \limits^{\sum {\,\, = \,\,360} } \,\,D = 90\,\,\,\,\, \Rightarrow \,\,\,\,\,ABCD\,\,\underline {{\text{any}}} \,\,{\text{rectangle}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{INSUFF}}.\]
\[\left( 2 \right)\,\,AB = AD\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{trivial}}\,\,{\text{geometric}}\,\,{\text{bifurcation}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{INSUFF}}.\]
\[\left( {1 + 2} \right)\,\,\,\left\{ {\,\left. \begin{gathered}
\,{\text{rectangle}}\,\,\,\, \Rightarrow \,\,\,\,{\text{parallelogram}} \hfill \\
\,\,\left[ {AB = AD} \right]\,\, \cap \,\,{\text{parallelogram}}\,\,\,\, \Rightarrow \,\,\,\,{\text{rhombus}} \hfill \\
\end{gathered} \right\}} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\text{SUFF}}{\text{.}}\]
\[\left( * \right)\,\,\,\left\{ \begin{gathered}
\,{\text{rectangle}} \hfill \\
\,{\text{rhombus}} \hfill \\
\end{gathered} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{square}}\]
This solution follows the notations and rationale (quadrilaterals properties) taught in the GMATH method.
Regards,
Fabio. 
