In figure, $\mathrm{P}$ is the point of contact of the tangent $\text{OP}$ with the circle.

If $\mathrm{m}(arc\mathrm{PR})=140$, $\angle \mathrm{POR}=36°$, find $\mathrm{m}(arc\mathrm{PQ})$.

In figure, $\mathrm{P}$ is the point of contact of the tangent $\mathrm{OP}$ to the circle.

If $\mathrm{OP}=7.2 \mathrm{units},\mathrm{OQ}=3.2 \mathrm{units}$, find $\mathrm{OR}$ and $\mathrm{QR}$.

In figure, $\mathrm{P}$ is the point of contact of the tangent $\mathrm{OP}$ with the circle.

If $\mathrm{OP}=7.2,\mathrm{OR}=16.2$ find $\mathrm{QR}$.

In figure, circles with centres $\mathrm{C}$ and $\mathrm{D}$ touch internally at point $\mathrm{E}$. $\mathrm{D}$ lies on the inner circle. Chord $\mathrm{EB}$ of the outer circle intersects inner circle at point $\mathrm{A}$. Prove that, $seg\mathrm{EA}\cong seg\mathrm{AB}$.

In figure, $\mathrm{seg} \mathrm{AB}$ is a diameter of a circle with centre $\mathrm{O}$. The bisector of $\angle \mathrm{ACB}$ intersects the circle at point $\mathrm{D}$ Prove that, $seg\mathrm{AD}\cong seg\mathrm{BD}$. Complete the following proof by filling in the blanks.

Proof: Draw seg $\mathrm{OD}$.

$\angle \mathrm{ACB}=$_____ angle inscribed in a semicircle.

$\angle \mathrm{DCB}=$_____ $\mathrm{CD}$ is the bisector of $\angle \mathrm{C}$

$\mathrm{m}(arcDB)=$_____ inscribed angle theorem 

$\angle \mathrm{DOB}=$_____ definition of measure of an arc $\left(\mathrm{I}\right)$

$seg\mathrm{OA}\cong seg\mathrm{OB}$_____ $\left(\mathrm{II}\right)$

$\therefore$ line $\mathrm{OD}$ is _____ of $segAB$ _____ from $\left(\mathrm{I}\right)$ and $\left(\mathrm{II}\right)$

$\therefore seg\mathrm{AD}\cong seg\mathrm{BD}$

In the figure, $\mathrm{seg} \mathrm{MN}$ is a chord of a circle with centre $\mathrm{O}.$ $\mathrm{MN}=25$ units. $\mathrm{L}$ is a point on chord $\mathrm{MN}$ such that $\mathrm{ML}=9$ units and $\mathrm{d}(\mathrm{O},\mathrm{L})=5$ units. Find the radius of the circle in units.

In figure, two circles intersect each other at points $\mathrm{S}$ and $\mathrm{R}$. Their common tangent $\mathrm{PQ}$ touches the circle at points $\mathrm{P},\mathrm{Q}$. Prove that,$\angle \mathrm{PRQ}+\angle \mathrm{PSQ}=180°$.

In figure, two circles intersect at points $\mathrm{M}$ and $\mathrm{N}$. Secants drawn through $\mathrm{M}$ and $\mathrm{N}$ intersect the circles at points $\mathrm{R},\mathrm{S}$ and $\mathrm{P}, \mathrm{Q}$ respectively.

Prove that seg $\mathrm{SQ} \parallel$ seg $\mathrm{RP}$.

In figure, two circles intersect each other at points $\mathrm{A}$ and $\mathrm{E}$. Their common secant through $\mathrm{E}$ intersects the circles at points $\mathrm{B}$ and $\mathrm{D}$. The tangents of the circles at points $\mathrm{B}$ and $\mathrm{D}$ intersects each other at point $\mathrm{C}$. Prove that $\square \mathrm{ABCD}$ is cyclic.