Question

Given figure $\text{ABCD}$ is a parallelogram, $\text{P}\text{and}\text{Q}$ are midpoints of sides $\text{AB}$and $\text{DC}$ respectively, then prove that$\text{APCQ}$ is a parallelogram.

Answer 1

$ABCD$ is a parallelogram.

$P,Q$ are mid-point of $AB$ and $CD$ respectively.

since $AB\parallel DC$ [ Opposite sides of parallelogram are parallel ]

$\Rightarrow$ $AP\parallel QC$ [ Parts of parallel lines are parallel ]

since $AB=CD$ [ Opposite sides of parallelogram are equal ]

$\Rightarrow$ $\frac{1}{2}AB=\frac{1}{2}CD$

$\Rightarrow$ $AP=QC$ [ $P$ is mid-point of $AB$ and $Q$ is mid-point of $DC$ ]

Since $AP\parallel QC$ and $AP=QC$
One pair of opposite sides of $APCQ$ is equal and parallel.

$\therefore$ $APCQ$ is a parallelogram.