Question : PQ is a tangent of a circle at T. If TR = TS where R and S are points on the circle and $\angle RST=65^{\circ}$, the $\angle PTS=?$
Option 1: $65^{\circ}$
Option 2: $130^{\circ}$
Option 3: $115^{\circ}$
Option 4: $55^{\circ}$
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Correct Answer: $115^{\circ}$
Solution :
In $\triangle TRS$
$TR=TS$
$\therefore \angle TRS= \angle RST$
Thus, $\angle TRS=65^\circ$
Now, $\angle STQ$ is the angle between the tangent $PTQ$ and chord $TS$
$\therefore \angle STQ=\angle TRS = 65^\circ$
Tangent $PTQ$ is a straight line,
$\therefore \angle PTS+\angle STQ=180^\circ$
⇒ $\angle PTS+65^\circ=180^\circ$
⇒ $\angle PTS=180^\circ-65^\circ = 115^\circ$
Hence, the correct answer is $115^\circ$.
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