Question

$ABCD$ is a quadrilateral in which $\overline{AB}=\overline{CD}$ and $\overline{AD}=\overline{BC}$. Show that it is a parallelogram.

Answer 1

REF.image.

Given:In $\square ABCD$

$AB=DCandAD=BC$

To prove: ABCD is a parallelogram

Construction: Draw diagonal AC and DB.

Proof: in $△ABC$ and $△ADC$

$AD=BC$ [Given]

$AB=DC$ [Given]

$AC=AC$ [Common side]

$\therefore$ By $SSS$ property

$△ADC\cong △ACB$

$\therefore \angle DAC=\angle DCA$

$\therefore AB||DC$ [By theorem]

In$△ABDand△DCB$

DB=DB [Common side]

AD=BC [Given]

AB=DC [Given]

$\therefore △ABD\cong △DCB$

$\therefore AD||BC$

Since $AB||DC$ and $AD||BC$.

$△ABCD$ is parallelogram