$ABC$ is an isosceles triangle in which $AB=AC$, $AD$ bisects the exterior angle $PAC$ and $CD\parallel BA$. Prove that $\angle DAC=\angle BCA$

$ABC$ is an isosceles triangle in which $AB=AC$, $AD$ bisects the exterior angle $PAC$ and $CD\parallel BA$. Prove that quadrilateral $ABCD$ is a parallelogram.

$ABCD$ is a parallelogram and $BD$ is one of its diagonals. $AP$ and $CQ$ are perpendiculars on $BD$ from the vertices $A$ and $C$ respectively. Prove that $△APB\cong △CQD$

$ABCD$ is a parallelogram and $BD$ is one of its diagonals. $AP$ and $CQ$ are perpendiculars on $BD$ from the vertices $A$ and $C$ respectively. Prove that $AP=CQ$.

$ABCD$ is a rectangle in which diagonal $AC$ bisects both the angles $A$ and $C$. Then prove that $ABCD$ is a square.

$ABCD$ is a rectangle in which diagonal $AC$ bisects both the angles $A$ and $C$. Then prove that diagonal $BD$ bisects both angles $B$ and $D$.

$∆ABC$ and $∆DEF$ are such that $AB$ and $BC$ are equal and parallel to $DE$ and $EF$ respectively. Prove that $AC$ and $DF$ are equal and parallel.

$ABCD$ is a trapezium where $AB\parallel DC$ . $E$ is the midpoint of $AD$. A line drawn from $E$, parallel to $AB$, meets $BC$ at point $F$. Prove that $F$ is the mid point of $BC$.

$ABCD$ is rhombus and $P, Q, R, S$ are the midpoints of the sides $AB, BC, CD$ and $DA$ respectively. Show that quadrilateral $PQRS$ is a rectangle.