Properties of a Parallelogram

$\mathrm{ABCD}$ is a rhombus in which altitude from $\mathrm{D}$ to side $\mathrm{AB}$ bisects $\mathrm{AB}$. Find the angles of the rhombus.

The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is $60°$. Find the angles of the
parallelogram.

Diagonals $\mathrm{AC}$ and $\mathrm{BD}$ of a parallelogram $\mathrm{ABCD}$ intersect each other at $\mathrm{O}$. If $\mathrm{OA}=3 \mathrm{cm}$ and $\mathrm{OD}=2 \mathrm{cm},$ determine the lengths of $\mathrm{AC}$ and $\mathrm{BD}$.

In the given figure, $\mathrm{ABCD}$ and $\mathrm{AEFG}$ are two parallelograms.

If $\angle \mathrm{C}=55°$ and $\angle \mathrm{F}=k°$, then find the value of $k$.

Opposite angles of a quadrilateral $\mathrm{ABCD}$ are equal. If $\mathrm{AB}=4 \mathrm{cm}$ and $\mathrm{CD}=x \mathrm{cm}$, then find the value of $x$.

Diagonals of a quadrilateral $\mathrm{ABCD}$ bisect each other. If $\angle \mathrm{A} = 35°$ and $\angle \mathrm{B}=k°$, then find the value of $k$.

$D, E$ and $F$ are respectively the mid-points of the sides $AB, BC$ and $CA$ of a triangle $ABC$. Prove that by joining these mid-points $D, E$ and $F$, the triangle $ABC$ is divided into four congruent triangles.

$P$ and $Q$ are points on opposite sides $AD$ and $BC$ of a parallelogram $ABCD$ such that $PQ$ passes through the point of intersection $O$ of its diagonals $\mathrm{AC}$ and $\mathrm{BD}$. Show that $\mathrm{PQ}$ is bisected at $\mathrm{O}$.

In the given figure, $AB\Vert DE, AB=DE, AC\Vert DF$ and $AC=DF$. Prove that $\mathrm{BC}\Vert \mathrm{EF}$ and $\mathrm{BC}=\mathrm{EF}$.

$P$ and $Q$ are the midpoints of the opposite sides $AB$ and $CD$ of a parallelogram $ABCD$. $AQ$ intersects $DP$ at $S$ and $BQ$ intersects $CP$ at $R$. Show that $PQRS$ is a parallelogram.

A diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus.

In the given figure, $\mathrm{P}$ is the midpoint of side $\mathrm{BC}$ of a parallelogram $\mathrm{ABCD}$ such that $\angle \mathrm{BAP}=\angle \mathrm{DAP}$. Prove that $\mathrm{AD}=2\mathrm{CD}$.

Through $\mathrm{A}, \mathrm{B}$ and $\mathrm{C},$ lines $\mathrm{RQ}, \mathrm{PR}$ and $\mathrm{QP}$ have been drawn, respectively parallel to sides $\mathrm{BC}, \mathrm{CA}$ and $\mathrm{AB}$ of a $△\mathrm{ABC}$ as shown in the given figure. Show that $\mathrm{BC}=\frac{1}{2}\mathrm{QR}$.

$\mathrm{E}$ and $\mathrm{F}$ are points on diagonal $\mathrm{AC}$ of a parallelogram $\mathrm{ABCD}$ such that $\mathrm{AE}=\mathrm{CF}$. Show that $\mathrm{BFDE}$ is a parallelogram.

In the given figure, it is given that $\mathrm{BDEF}$ and $\mathrm{FDCE}$ are parallelograms. Can you say that $\mathrm{BD}=\mathrm{CD}$? Why or why not?

Diagonals of a rectangle are equal and perpendicular. Is this statement true? Give reason for your answer.

All the angles of a quadrilateral are equal. What special name is given to this quadrilateral?

Diagonals of a parallelogram are perpendicular to each other. Is this statement true? Give reason for your answer.

Which of the following is not true for a parallelogram?

The figure formed by joining the mid-points of the sides of a quadrilateral $\mathrm{ABCD}$, taken in order, is a square only, if