91Ó°ÊÓ

Given:

$TR$ and $TS$ are tangents to the circle at $O$ from $T$.

$m∠RTS=36°$.

$TR=TS$

In $\mathrm{Δ}RTS$,

$m∠RTS=36°$.

Since $TR=TS$,

$∠TSR=∠TRS=x$ (angles opposite to equal sides are equal).

Again,

since $m∠RTS+m∠TRS+m∠TSR={180}^0$

$180{}^{∘}=36{}^{∘}+x+x$,

$180{}^{∘}=36{}^{∘}+2x$,

$x=72°$

$∴m∠TSR=m∠TRS=72°$.

Therefore,the measures are: $m∠TSR=72°$ and $m∠TRS=72°$.

Given:

TR and $TS$are tangents to the circle at $O$from $T$.

$m∠RTS=36°$.

In quadrilateral $RTSO$,

$∠RTS+∠TSO+∠SOR+∠ORT=360°$

$∠TRO=∠TSO=90°$ (a tangent form an angle of $90°$ with the surface of the circle.)

we have,

$36{}^{∘}+90{}^{∘}+90{}^{∘}+∠ROS=360°$

$216{}^{∘}+∠ROS=360°$

$m∠ROS=144°$

Now in $\mathrm{Δ}SOR$,

$RO=SO$ (radii of the circle),

$∠ORS=∠OSR=x$ (angle opposite to equal sides are equal).

$∠ROS+∠ORS+∠OSR=180°$

$144{}^{∘}+x+x=180°$,

$2x=36°$

$x=18°$

Therefore, value of, $m∠ORS=m∠OSR=18°$.

Given:

$TR$and $TS$are tangents to the circle at $O$from T.

$m∠RTS=36°$.

In quadrilateral $RTSO$,

$∠RTS+∠TSO+∠SOR+∠ORT=360°$

$∠TRO=∠TSO=90°$ (a tangent form an angle of $90°$ with the surface of the circle.)

we have,

$36{}^{∘}+90{}^{∘}+90{}^{∘}+∠ROS=360°$,

$216{}^{∘}+∠ROS=360°$,

$m∠ROS=144°$.

Therefore, the value of $m∠ROS=144{}^{∘}.$