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.

$B$ and $D$ are the midpoints of $AE$ and $GF$ of the parallelogram $AEFG$. Prove that $ABGD$ is a parallelogram.

The points $E, F, G$ and $H$ are on the sides $AB, BC, CD$ and $DA$ respectively of the rectangle $ABCD$, such that $AE=BF=CG=DH$. Prove that $EFGH$ is a parallelogram.

Two squares $ABPQ$ and $ADRS$ are constructed on the sides $AB$ and $AD$ respectively of the parallelogram $ABCD$ on the outside of $ABCD$. Prove that $∆PRC$ is isosceles.

$\angle BAD$ of the parallelogram $ABCD$ is an obtuse angle. Two equilateral triangles $△ABP$ and $△ADQ$ are constructed on the sides $AB$ and $AD$ on the outside of $ABCD$. Prove that $△CPQ$ is an equilateral triangle.