Question

In parallelogram, both pairs of opposite sides are ....................

• A. equal
• B. parallel and equal
• C. perpendicular
• D. perpendicular and equal

Answer 1

Answer: (B)

parallel and equal

In $ABCD$ is a parallelogram,

$\Rightarrow$ We know that $AB\parallel CD$

Here $AC$ is a transversal for the parallel lines $AB$ and $CD$

$\Rightarrow$ So, $\angle BAC=\angle DCA$ [Alternate interior angles are equal] $...\left(1\right)$

$\Rightarrow$ Similarly, we know that $BC\parallel AD$

$\Rightarrow \angle BCA=\angle DAC$ $...\left(2\right)$

In $△ABC$ and $△ADC,$

We have $\angle BAC=\angle DCA$ [From ( 1 )]

$\Rightarrow$ $AC$ is the common side

$\Rightarrow \angle BCA=\angle DAC$ [From ( 2 )]

$\Rightarrow$ So, $△ABC\cong △CDA$ [As per ASA Congruence rule]

$\therefore AB=CD$ and $BC=AD$ [Corresponding sides of congruent triangle are equal ]

Hence, both pairs of opposite sides in a parallelogram are parallel and equal.