Angle Subtended by a Chord at a Point

In the figure, if $\angle ABC=50°$ and $\angle BDC=40°$, calculate $\angle BCA$.

In the figure, if $\angle ABC=50°$ and $\angle BDC=40°$, calculate $\angle BAC$.

In the adjoining figure, $O$ is the centre of the circle. $m\left(arc XYB\right)=55°$. Find the measure of the arc $ADX$ (in degrees).

In Fig, $ABCD$ is a cyclic quadrilateral in which $AC$ and $BD$ are its diagonals. If $\angle DBC=55°$ and $\angle BAC=45°$, find $\angle BCD$.

In the given figure $\text{'}O\text{'}$ is the centre of the circle and $AB, CD$ are equal chords. If $\angle AOB=70°$. If $\angle OCD=k°$, then find the value of $k$.

The given figure shows a circle with centre $\mathrm{O}$. Also, $\mathrm{PQ}=\mathrm{QR}=\mathrm{RS}$ and $\angle \mathrm{POS}=150°$. If $\angle \mathrm{PQR}=k°$, then find the value of $k$.

A regular polygon is inscribed in a circle. If each side of a regular polygon subtends an angle $60°$ at the center of the circle, then what is the number of sides in the regular polygon?

The point $O$ is the center of the circle. A square $ABCD$ is drawn inside the circle. If the side of the square subtends an angle of

$k°$ at the center of the circle then find $k$.

and

In the above figure $O$ is the centre of the circle. If $\angle AOB=\angle COD$ and chord $AB=2.5 \mathrm{cm}$, then find the length of chord $CD$. (Write your answer without unit)

$\mathrm{PT}$ is a tangent to the circle at $\mathrm{T}.$ If $\angle \mathrm{ABC} = 70°$ and $\angle \mathrm{ACB} = 50°,$ calculate $\angle \mathrm{BAT}$.

$\mathrm{PT}$ is a tangent to the circle at $\mathrm{T}.$ If $\angle \mathrm{ABC} = 70°$ and $\angle \mathrm{ACB} = 50°,$ calculate $\angle \mathrm{CBT}$.

In the given figure, $\Delta ABC$ and $\Delta DBC$ are inscribed in a circle such that $\angle BAC=60°$ and $\angle DBC=$$50°.$Then $\angle BCD$ equal to

Equal chords of a circle subtend equal angles at

In the following figure, $\mathrm{PQ}$ is the tangent to the circle at $\mathrm{A}, \mathrm{DB}$ is the diameter and $\mathrm{O}$ is the centre of the circle. If $\angle \mathrm{ADB}=30°$ and $\angle \mathrm{CBD}=60°,$ calculate $\angle \mathrm{CDB}.$

Bisectors of vertex angles $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ of a triangle $\mathrm{ABC}$ intersect its circumcircle at the points $\mathrm{D}, \mathrm{E}$ and $\mathrm{F}$ respectively. Prove that angle $\mathrm{EDF}=90°-\frac{1}{2}\angle \mathrm{A}.$

In the given figure, $\mathrm{PQ}$ is a tangent to the circle at $\mathrm{A}. \mathrm{AB}$ and $\mathrm{AD}$ are bisectors of $\angle \mathrm{CAQ}$ and $\angle \mathrm{PAC}.$ If $\angle \mathrm{BAQ} = 30°,$ prove that $\mathrm{BD}$ is diameter of the circle.
 

In the above figure $O$ is the centre of the circle. Three chords $AB,PQ$ and $CD$ of this circle are drawn in such a way that $AB=PQ=CD$ if $\angle AOB=65°$ then what will be the values of $\angle POQ$ and $\angle COD$ ?

In the figure, $BC$ is the diameter and $\angle BAO = {60}^{\circ }$ then $\angle ADC$ is equal to

In the figure, if $\angle DAB=60°$ and $\angle ABD=50°$, then $\angle ACB$ is equal to

State true or false for given statement and also write the justification for your answer.

In figure, $\angle CAD=\angle CED$.