Prove that if two opposite angles are equal and two opposite sides of any quadrilateral be parallel, then it is parallelogram.

Prove that if the diagonals of any parallelogram bisect each other orthogonally, then it is a rhombus.

The diagonal $SQ$ of the parallelogram $PQRS$ is divided into three equal parts at $K$ and $L$. $PK$ interscct $SQ$ at $M$ and $RL$ intersect $PQ$ at $N$. Prove that $PMRN$ is a parallelogram.

Prove that the quadrilateral obtained by successively joining the midpoints of a parallelogram is also a parallelogram.

$ABCD$ is a rectangle. Two equilateral triangles $△ABP$ and $△CDQ$ are constructed taking two opposite sides $AB$ and $CD$ as their sides on the outside of the rectangle $ABCD$. Prove that $APCQ$ is a parallelogram.

$E$ and $F$ are the midpoints of $AB$ and $AC$ respectively of $△ABC$. The straight line $EF$ is extended to the point $D$ such that $EF=FD$. Prove that $ADCE$ is a parallelogram.

The diagonals of the square $ABCD$ intersect each other at $O$. Let $OP\perp AB$ is constructed. Prove that $\angle AOP=\angle PBO$.

In the quadrilateral $ABCD, AD=BC$ and $\angle BAD=\angle ABC$. Prove that the quadrilateral is an isosceles trapezium.

$ABCD$ is a quadrilateral. Two parallelograms $BADQ$ and $ADCP$ are constructed. Prove that $PQ$ and $BC$ bisects each other equally.