In parallelogram $ABCD$, two points $P$ and $Q$ are taken on diagonal $BD$ such that $DP = BQ$ (see Fig. 8.12). Show that: (i) $\Delta APD \cong \Delta CQB$

           

$ABCD$ is a trapezium in which $AB \| CD$ and $AD = BC$ (see Fig. 8.14). Show that (i) $\angle A = \angle B$ [Hint : Extend $AB$ and draw a line through $C$ parallel to $DA$ intersecting $AB$ produced at $E$.]

$ABCD$ is a rectangle in which diagonal $AC$ bisects $\angle A$ as well as $\angle C$. Show that: (ii) diagonal $BD$ bisects $\angle B$ as well as $\angle D$.

$ABCD$ is a quadrilateral in which $P, Q, R$ and $S$ are mid-points of the sides $AB, BC, CD$ and $DA$ (see Fig 8.20). $AC$ is a diagonal. Show that : (ii) $PQ = SR$

$ABCD$ is a trapezium in which $AB \| CD$ and $AD = BC$ (see Fig. 8.14). Show that (ii) $\angle C = \angle D$ [Hint : Extend $AB$ and draw a line through $C$ parallel to $DA$ intersecting $AB$ produced at $E$.]

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

$ABCD$ is a quadrilateral in which $P, Q, R$ and $S$ are mid-points of the sides $AB, BC, CD$ and $DA$ (see Fig 8.20). $AC$ is a diagonal. Show that : (iii) $PQRS$ is a parallelogram.

$ABCD$ is a trapezium in which $AB \| CD$ and $AD = BC$ (see Fig. 8.14). Show that (iii) $\Delta ABC \cong \Delta BAD$ [Hint : Extend $AB$ and draw a line through $C$ parallel to $DA$ intersecting $AB$ produced at $E$.]

$ABCD$ is a trapezium in which $AB \| CD$ and $AD = BC$ (see Fig. 8.14). Show that (iv) diagonal $AC$ = diagonal $BD$ [Hint : Extend $AB$ and draw a line through $C$ parallel to $DA$ intersecting $AB$ produced at $E$.]

In parallelogram $ABCD$, two points $P$ and $Q$ are taken on diagonal $BD$ such that $DP = BQ$ (see Fig. 8.12). Show that: (iv) $AQ = CP$

      

In parallelogram $ABCD$, two points $P$ and $Q$ are taken on diagonal $BD$ such that $DP = BQ$ (see Fig. 8.12). Show that: (v) $APCQ$ is a parallelogram

      

If the diagonals of a parallelogram are equal, then show that it is a rectangle.

$ABCD$ is a parallelogram and $AP$ and $CQ$ are perpendiculars from vertices $A$ and $C$ on diagonal $BD$ (see Fig. 8.13). Show that (i) $\Delta APB \cong \Delta CQD$

$ABCD$ is a parallelogram and $AP$ and $CQ$ are perpendiculars from vertices $A$ and $C$ on diagonal $BD$ (see Fig. 8.13). Show that (ii) $AP = CQ$

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

$ABCD$ is a trapezium in which $AB \| DC$, $BD$ is a diagonal and $E$ is the mid-point of $AD$. A line is drawn through $E$ parallel to $AB$ intersecting $BC$ at $F$ (see Fig. 8.21). Show that $F$ is the mid-point of $BC$.

In a parallelogram $ABCD$, $E$ and $F$ are the mid-points of sides $AB$ and $CD$ respectively (see Fig. 8.22). Show that the line segments $AF$ and $EC$ trisect the diagonal $BD$.

Show that the diagonals of a square are equal and bisect each other at right angles.

Diagonal $AC$ of a parallelogram $ABCD$ bisects $\angle A$ (see Fig. 8.11). Show that (i) it bisects $\angle C$ also,

$ABC$ is a triangle right angled at $C$. A line through the mid-point $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that (ii) $MD \perp AC$