Q.

$ABCD$ 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

Q. In given figure $ABCD$ is a quadrilateral in which $P,Q,R$ and $S$ are mid-points of the sides. $AB,BC,CD$ and $DA$. $AC$ is a diagonal. Show that:
$SR\parallel AC$ and $SR=\frac{1}{2}AC$

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

Q. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Q. In parallelogram $ABCD$, two point $P$ and $Q$ are taken on diagonal $BD$ such that $DP=BQ$. Show that
(i) $△APD\cong △CQB$
(ii) $AP=CQ$
(iii) $△AQB\cong △CPD$
(iv) $AQ=CP$
(v) $APCQ$ is a parallelogram

Q. $ABCD$ is a parallelogram and $AP$ and $CQ$ are perpendiculars from vertices $A$ and $C$ on diagonal $BD$ Show that
i) $△APB\cong △CQD$
ii) $AP=CQ$

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

Q. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

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

Q. $ABCD$ is a rhombus. Show that diagonal $AC$ bisects $\angle A$ as well as $\angle C$ and diagonal $BD$ bisects $\angle B$ as well as $\angle D$.

Q. Diagonal $AC$ of a parallelogram $ABCD$ bisects $\angle A$. Show that
(i) It bisects $\angle C$ also,
(ii) $ABCD$ is a rhombus.

Q. $ABC$ is a triangle right angled at $C$. A line through the mid-point $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ and $D$. Show that
(i) $D$ is the mid-point of $AC$
(ii) $MB\perp AC$
(iii) $CM=MA=\frac{1}{2}AB$

Q. In given figure $ABCD$ is a trapezium in which $AB\parallel CD$ and $AD=BC$. Show that
(i) $\angle A=\angle B$
(ii) $\angle C=\angle D$
(iii) $△ABC\cong △BAD$
(iv) diagonal $AC=$ diagonal $BD$

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

Q. The angles of quadrilateral are in the ratio $3:5:9:13$. Find all the angles of the quadrilateral.

Q. $ABCD$ is a trapezium in which $AB\parallel DC,BD$ 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). Show that $F$ is the mid-point of $BC$.

Q. Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other

Q. In $△ABC$ and $△DEF,AB=DE,AB\parallel DE,BC=EF$ and $BC\parallel EF$. Vertices $A,B$ and $C$ are joined to vertices $D,E$ and $F$ respectively. Show that
(i) Quadrilateral $ABED$ is a parallelogram
(ii) Quadrilateral $BEFC$ is a parallelogram
(ii) $AD\parallel CF$ and $AD=CF$
(iv) Quadrilateral $ACFD$ is a parallelogram
(v) $AC=DF$
(vi) $△ABC\cong △DEF$

Q. $ABCD$ is a rectangle 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 rhombus.