Cyclic Quadrilaterals

$PQRS$ is a cyclic quadrilateral such that $PR$ is a diameter of the circle. If $\angle QPR=67°$ and $\angle SPR=72°$, $\angle QRS=$

$ABCD$ is a  cyclic quadrilateral where $\angle DBC=27°, \angle ABD=50°$ and $\angle ADB=33°$, then find $\angle ADC$ in degree $\left(°\right)$.

Prove that any non-isosceles trapezium is not cyclic.

In the given figure, $\mathrm{C}$ and $\mathrm{D}$ are points on the semi-circle described on $\mathrm{AB}$ as diameter. Given that, angle $\mathrm{BAD} = 70°$ and angle $\mathrm{DBC} = 30°$, then calculate angle $\mathrm{BDC}.$

If the given figure, $\mathrm{O}$ is the centre of the circle. The tangents at $\mathrm{B}$ and $\mathrm{D}$ intersect each other at point $\mathrm{P}.$ If $\mathrm{AB}$ is parallel to $\mathrm{CD}$ and $\angle \mathrm{ABC}=55º,$ find $\angle \mathrm{BPD}$.

$\mathrm{ABCDE}$ is a cyclic pentagon with centre of its circumcircle at point $\mathrm{O}$ such that $\mathrm{AB}=\mathrm{BC}=\mathrm{CD}$ and angle $\mathrm{ABC}=120°$. Calculate $\angle \mathrm{BED}$.

Consider the given figure.

The measure of ∠ L + ∠ M + ∠ N + ∠ P = _____ degrees.

In the given figure, P, Q, R, S and T are points lying on the circumference of the circle. O is the centre and PT is the diameter of the circle.

The value of ∠ PQR + ∠RST = _____ degrees.

If O is the centre of the circle and side PQ is produced to R, then the measure of ∠ SQR = _____ degrees.

In the given figure, $ ∆$ IJK is an isosceles triangle with IJ = IK.

The measure of ∠ JLK = _____ degrees.

In the figure given, O is the centre of the circle and L, M, N and P are points on the circumference of the circle. LP is the diameter of the circle.

The measure of ∠ PLN = _____ degrees.

In the figure given, O is the centre of the circle and P, Q and R are points lying on the circumference of the circle.

The measure of ∠ PRQ = _____ degrees.

In the figure below, points $A, B, C$ and $D$ lie on the circle. The measures of $\angle AEC$ and $\angle CBE$ are ${45}^o$ and ${85}^o$ respectively.

The measure of $\angle BAD =$ _____ degrees.

Two circles intersect in point $\mathrm{P}$ and $\mathrm{Q}$. A secant passing through $\mathrm{P}$ intersects the circles in $\mathrm{A}$ and $\mathrm{B}$ respectively. Tangents to the circles at $\mathrm{A}$ and $\mathrm{B}$ intersect at $\mathrm{T}$. Prove that $\mathrm{A},\mathrm{Q},\mathrm{B}$ and $\mathrm{T}$ lie on a circle.

In the given figure, $\mathrm{PAT}$ is tangent to the circle with centre $\mathrm{O}$, at point $\mathrm{A}$ on its circumference and is parallel to Chord $\mathrm{BC}$. If $\mathrm{CDQ}$ is a line segment , show that $\angle \mathrm{BAP}=\angle \mathrm{ADQ}$.

In a square $\mathrm{ABCD}$, its diagonals $\mathrm{AC}$ and $\mathrm{BD}$ intersect each other at point  $\mathrm{O}$. The bisector of angle $\mathrm{DAO}$ meets $\mathrm{BD}$ at point $\mathrm{M}$ and the bisector of angle $\mathrm{ABD}$ meets $\mathrm{AC}$ at $\mathrm{N}$ and $\mathrm{AM}$ at $\mathrm{L}$. show that :

$\mathrm{ALOB}$ is a cyclic quadrilateral.

Two circles intersect each other at points $\mathrm{A}$ and $\mathrm{B}.$ A straight line $\mathrm{PAQ}$ cuts the circles at $\mathrm{P}$ and $\mathrm{Q}.$ If the tangents at $\mathrm{P}$ and $\mathrm{Q}$ intersect at point $\mathrm{T},$ show that the points $\mathrm{P},\mathrm{B},\mathrm{Q}$ and $\mathrm{T}$ are concyclic.

Prove that any four vertices of a regular pentagon are con-cyclic (lie on the same circle).