Diagonals of a parallelogram intersect each other at point $\mathrm{O}$. If $AO= 5$, $BO=12$ and $AB=13$ then, show that quadrilateral $ABCD$ is a rhombus.

In the figure, $\mathrm{PQRS}$ and $\mathrm{ABCR}$ are two parallelograms. If $\angle \mathrm{P} = 110°$ then, find the measures of all angles of $\mathrm{ABCR}$.

In figure, $\mathrm{ABCD}$ is a parallelogram. Point$\mathrm{E}$ is on the ray $\mathrm{AB}$ such that $\mathrm{BE}=\mathrm{AB}$, then prove that line $\mathrm{ED}$ bisects segment $\mathrm{BC}$ at point $\mathrm{F}$.

In figure, $\mathrm{ABCD}$ is a parallelogram, $\mathrm{P}$ and $\mathrm{Q}$ are midpoints of side $\mathrm{AB}$ and $\mathrm{DC}$ respectively, then prove $\mathrm{APCQ}$ is a parallelogram.

Using opposite angles test for parallelogram, prove that every rectangle is a parallelogram.

In figure, $\mathrm{G}$ is the point of concurrence of medians of $∆\mathrm{DEF}$. Take point $\mathrm{H}$ on ray such that $\mathrm{D}-\mathrm{G}-\mathrm{H}$ and $\mathrm{DG}=\mathrm{GH}$, then prove that $\mathrm{GEHF}$ is a parallelogram.

Prove that quadrilateral formed by the intersection of angle bisectors of all angles of a parallelogram is a rectangle.

In figure, if points $\mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S}$ are on the sides of parallelogram such that $\mathrm{AP}=\mathrm{BQ}=\mathrm{CR}=\mathrm{DS}$, then prove that $\mathrm{PQRS}$ is a parallelogram.

Diagonals of a rectangle ABCD intersect at point O. If AC = $8$ cm then find BO and if $\angle CAD=35°$ then $\angle ACB$ is $k°$. Find the value of $k$.