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Is quadrilateral ABCD a square?

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Is quadrilateral ABCD a square?

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Is quadrilateral ABCD a square?

(1) A = B = C = 90º

(2) AB = AD

OA C

Source: Magoosh

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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.

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BTGmoderatorDC wrote:
Is quadrilateral ABCD a square?

(1) A = B = C = 90º

(2) AB = AD

OA C

Source: Magoosh
Question: Is quadrilateral ABCD a square?

Let's take each statement one by one.

(1) A = B = C = 90º

A quadrilateral has four angles. Since each of the three angles is equal to 90º, we have the fourth angle /_D = 90º

A quadrilateral having four equal angles can be either a rectangle or a square. No unique answer. Insufficient.

(2) AB = AD

=> Adjacent sides are equal.

The figure can be either a square or any quadrilateral. No unique answer. Insufficient.

(1) and (2) together

From Statement 2 AB = AD, we know that the adjacent sides are equal and from Statement 1, we know that the angle between them is 90º, thus, the other two sides must also be equal with each angle being 90º, making the quadrilateral a square. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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