Angle Relationships in Circles

If a line is drawn through an end point of a chord of a circle so that the angle formed with the chord is equal to the angle subtended by the chord in the alternate segment, then the line is a tangent to the circle.

If secants containing chords $PQ$ and $MN$ of a circle, intersect outside the circle at $E$. If $PE=6 \mathrm{cm}, ME=3 \mathrm{cm}$ and $NE=12 \mathrm{cm}$, the length of $QE$ is ___

In the figure, $PQRS$ is a cyclic quadrilateral. Find the values of $x$ and $y$.

In the figure, $PQRS$ is a cyclic quadrilateral. Find the values of $x$ and $y$.

In the following figure, $\mathrm{PQ}=\mathrm{QR},$ $\angle \mathrm{RQP}=68º,$ $\mathrm{PC}$ and $\mathrm{CQ}$ are tangents to the circle with centre $\mathrm{O}.$ Calculate the value of $\angle \mathrm{QCP}.$

In the given figure,$\mathrm{ABCD}$ is a cyclic quadrilateral, $\mathrm{PQ}$ is tangent to the circle at point $\mathrm{C}$ and $\mathrm{BD}$ is its diameter. If $\angle \mathrm{DCQ}=40°$ and $\angle \mathrm{ABD}=60°,$ find $\angle \mathrm{DBC}$.

If $\mathrm{PQ}$ is a tangent to the circle at $\mathrm{R},$ calculate $\angle \mathrm{PRS}.$ Given that, $\mathrm{O}$ is the centre of the circle and angle $\mathrm{TRQ} = 30°.$

According to figure answer the following questions:

(i) $\angle BAQ$ is an alternate segment of circle.

(ii) $\angle DAP$ is an alternate segment of circle.

(iii) If $B$ is joined with $C$ then $\angle ACB$ is equal to which angle?

(iv) $\angle ABD$ and $\angle ADB$ is equal to which angle? 

In the given circle with centre with centre $\mathrm{O}, \angle \mathrm{ABC}=100º,$ $\angle \mathrm{ACD}=40º$ and $\mathrm{CT}$ is a tagent to the circle at $\mathrm{C}.$ Find $\angle \mathrm{ADC}$ and $\angle \mathrm{DCT}.$

Two circles touch each other internally at a point $\mathrm{P}.$ A chord $\mathrm{AB}$ of the bigger circle intersects the other circle in $\mathrm{C}$ and $\mathrm{D}.$ Prove that $\angle \mathrm{CPA} =\angle \mathrm{DPB}.$

Two circles with centres $\mathrm{O}$ and $\mathrm{O}\text{'}$ are drawn to intersect each other at points $\mathrm{A}$ and $\mathrm{B}.$ Centre $\mathrm{O}$ of one circle lies on the circumference of the other circle and $\mathrm{CD}$ is drawn tangent as to the circle with centre $\mathrm{O}\text{'}$ at $\mathrm{A}.$ Prove that $\mathrm{OA}$ bisects angle $\mathrm{BAC}.$

According to figure, $PQ$ and $XY$ are parallel tangents. If $\angle QRT=30°$ then find the value of $\angle TSY$.

According to the figure, if $\angle BAC = 80°$ then find the value of $\angle BCP.$