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# Is quadrilateral ABCD a square?

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Senior Manager
Joined: 21 Oct 2013
Posts: 408

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05 Aug 2014, 10:38
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70% (00:33) correct 30% (00:39) wrong based on 416 sessions

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(1) AB=BC
(2) ABCD is a rectangle.
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Joined: 25 Apr 2012
Posts: 651
Location: India
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05 Aug 2014, 10:46
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goodyear2013 wrote:

(1) AB=BC
(2) ABCD is a rectangle.

From first we have ABCD is a square or Rhombus

From St 2, we have that Angles are at 90 deg

Combining we have AB=BC=CD=AD and all angles 90 deg.

Ans is C
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Joined: 22 Feb 2009
Posts: 153

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05 Aug 2014, 23:42
WoundedTiger wrote:
goodyear2013 wrote:

(1) AB=BC
(2) ABCD is a rectangle.

From first we have ABCD is a square or Rhombus

From St 2, we have that Angles are at 90 deg

Combining we have AB=BC=CD=AD and all angles 90 deg.

Ans is C

Yeah, (1) or (2) is insufficient
(1) and (2) is definitely a square !!!
Senior Manager
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28 Jan 2018, 23:14
1
WoundedTiger wrote:
goodyear2013 wrote:

(1) AB=BC
(2) ABCD is a rectangle.

From first we have ABCD is a square or Rhombus

From St 2, we have that Angles are at 90 deg

Combining we have AB=BC=CD=AD and all angles 90 deg.

Ans is C

hi

according to the statement 1, the quadrilateral does not have to be a rhombus also, because CD can be any straight line bigger than AB, and angle ADC can be an acute angle

taking 2 statements together we can see that the quadrilateral can only have 4 right angles, and so, CD cannot be larger than AB, and AB = BC

thanks, and cheers with kudos
Senior Manager
Joined: 27 Dec 2016
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29 Jan 2018, 19:17
Could anyone please explain why statement 2 is insufficient? Isn't there a property that says A rectangle can be a square as well?
Math Expert
Joined: 02 Sep 2009
Posts: 59587

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29 Jan 2018, 20:58
csaluja wrote:
Could anyone please explain why statement 2 is insufficient? Isn't there a property that says A rectangle can be a square as well?

A rectangle could be a square but it's not always a square. So, all squares are rectangles but not all rectangles are squares.

Check Properties of Polygons Questions from our Special Questions Directory.
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30 Jan 2018, 05:21
csaluja wrote:
Could anyone please explain why statement 2 is insufficient? Isn't there a property that says A rectangle can be a square as well?

hi

you can see Bunuel the great has already replied to your confusion, however, for some elementary clarification, remember that

rectangle means, 4 right angles
square means, 4 right angles + all sides are equal

now, hope this is clear that all squares possess all the characteristics of rectangles, but
rectangles do not possess all the characteristics of squares

thanks, and cheers with kudos if this is clear to you!
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
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22 Nov 2018, 18:47
goodyear2013 wrote:

(1) AB=BC
(2) ABCD is a rectangle.

$$ABCD\,\,\mathop = \limits^? \,\,{\text{square}}$$

$$\left( 1 \right)\,\,AB = BC\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{trivial}}\,\,{\text{geometric}}\,\,{\text{bifurcation}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{INSUFF}}.$$

$$\left( 2 \right)\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,{\text{ABCD}}\,\,{\text{rectangle}}\,\,{\text{non - square}} \hfill \\ \,{\text{Take}}\,\,{\text{ABCD}}\,\,\left( {{\text{rectangle}}\,\,{\text{and}}} \right)\,\,{\text{square}} \hfill \\ \end{gathered} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{INSUFF}}.$$

$$\left( {1 + 2} \right)\,\,\,\left\{ {\,\left. \begin{gathered} \,{\text{rectangle}}\,\,\,\, \Rightarrow \,\,\,\,{\text{parallelogram}} \hfill \\ \,\,\left[ {AB = BC} \right]\,\, \cap \,\,{\text{parallelogram}}\,\,\,\, \Rightarrow \,\,\,\,{\text{rhombus}} \hfill \\ \end{gathered} \right\}} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\text{SUFF}}{\text{.}}$$

$$\left( * \right)\,\,\,\left\{ \begin{gathered} \,{\text{rectangle}} \hfill \\ \,{\text{rhombus}} \hfill \\ \end{gathered} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{square}}$$

This solution follows the notations and rationale (quadrilaterals properties) taught in the GMATH method.

Regards,
Fabio.
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