$\mathrm{ABCD}$ is a $\parallel \mathrm{gm}$. $\mathrm{ABML}$ and $\mathrm{ADXY}$ are squares. Prove that $∆\mathrm{CXM}$ is isosceles.

$\mathrm{ABCD}$ and $\mathrm{ALMN}$ are squares. Prove that $\mathrm{BL}=\mathrm{DN}$.

Prove that the bisectors of any two adjacent angles of a parallelogram are at right angles.

In figure, $\mathrm{PQRS}$ is a parallelogram with bisectors $\mathrm{PA}, \mathrm{QD}, \mathrm{RC}$ and $\mathrm{SB}$ respectively of angles $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$. Show that $\mathrm{ABCD}$ is a rectangle.

In the figure, $\mathrm{ABCD}$ is a parallelogram and $\mathrm{X}$ is the midpoint of $\mathrm{BC}$. The line $\mathrm{AX}$ produced meets $\mathrm{DC}$ produced at $\mathrm{Q}$. The parallelogram is  $\mathrm{ABPQ}$ completed. Prove that: $∆\mathrm{ABX}\cong ∆\mathrm{QCX}$

In the figure, $\mathrm{ABCD}$ is a parallelogram and $\mathrm{X}$ is the midpoint of $\mathrm{BC}$. The line $\mathrm{AX}$ produced meets $\mathrm{DC}$ produced at $\mathrm{Q}$. The parallelogram $\mathrm{ABPQ}$ is completed. Prove that: $\mathrm{DC}=\mathrm{CQ}=\mathrm{QP}$

$\mathrm{ABCD}$ is a rhombus with $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ as midpoints of $\mathrm{AB}, \mathrm{BC}$ and $\mathrm{CD}$. Prove that $\mathrm{PQ}\perp \mathrm{QR}$.