In the quadrilateral ABCD below, the measure of $\angle ADC = 80°$. A teacher showed the quadrilateral to $4$ students and asked them to identify the combination of $3$ or more points through which a circle can be drawn.

Here are the responses from the students.

Anup : “$1$ circle only, through points $\mathrm{A}-\mathrm{O}-\mathrm{B}"$

Deepa: “$4$ circles only, through the set of points, $\mathrm{A}-\mathrm{O}-\mathrm{B}, \mathrm{B}-\mathrm{O}-\mathrm{C}, \mathrm{C}-\mathrm{O}-\mathrm{D}$ and $\mathrm{A}-\mathrm{O}-\mathrm{D}$.$\mathrm{A}-\mathrm{B}-\mathrm{C}-\mathrm{D}$"

Varun: “$1$ circle only, through the set of points ."

Ram: “$8$ circles, $\mathrm{A}-\mathrm{O}-\mathrm{B}, \mathrm{B}-\mathrm{O}-\mathrm{C}, \mathrm{C}-\mathrm{O}-\mathrm{D}, \mathrm{A}-\mathrm{O}-\mathrm{D}, \mathrm{A}-\mathrm{B}-\mathrm{C}, \mathrm{B}-\mathrm{C}-\mathrm{D}, \mathrm{C}-\mathrm{D}-\mathrm{A}$ and $\mathrm{D}-\mathrm{A}-\mathrm{B}$."

Who is correct?

Construct a cyclic quadrilateral $PQRS$ with $PQ=4 \mathrm{cm}, QR=6 \mathrm{cm}, PR=7.5 \mathrm{cm}, QS=7 \mathrm{cm}$.

$PQRS$ is a cyclic quadrilateral. Given $\angle QPS = 73°, \angle PQS = 55°$ and $\angle PSR=82°$ calculate $\angle RQS$.

Using ruler and compass only, construct a $∆ABC$ such that $BC=5 \mathrm{cm}$ and $AB=6.5 \mathrm{cm}$ and $\angle ABC=120°$. Construct a cyclic quadrilateral $ABCD$, such that $D$ is equidistant from $AB$ and $BC$.

$O$ is the centre of a circle and $AB$ a diameter. $ABCD$ is a cyclic quadrilateral. $\angle ABC=65°$,  $\angle DAC=40°$ then the measure of $\angle BCD$ is 

$\mathrm{PQRS}$ is a cyclic quadrilateral. Given $\angle \mathrm{QPS}=73°$, $\angle \mathrm{PQS}=55°$ and $\angle \mathrm{PSR}=82°$, calculate $\angle \mathrm{QRS}$.

In the figure, $ABCD$ is a cyclic quadrilateral. Also, $\angle A+\angle D=210°, \angle D+\angle C=250°$. What is $\angle A+\angle C$?

In the figure, if $\angle \mathrm{A}={100}^{\circ }$ then $\angle \mathrm{C}=$?

In the given figure $O$ is the centre of the circle, $\angle AOC=80°$

What is the measure of $\angle ADC$?

In the figure, $ABCD$ is a cyclic quadrilateral. Also, $\angle A+\angle D=210°, \angle D+\angle C=250°$. Find the measures of $\angle A$ and $\angle C$.

If two sides of a cyclic quadrilateral are parallel, then prove that other sides are equal and its diagonals are also equal to each other.

In the figure, the chord $\mathrm{BD}$ is perpendicular to the diameter $\mathrm{AC}$. If the measure of $\angle BCD$ is $k°$ then find the value of $k$.