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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Question : In $\triangle P Q R, S$ is a point on the side QR such that $\angle Q P S=\frac{1}{2} \angle P S R, \angle Q P R=78^{\circ}$ and $\angle P R S=44^{\circ}$. What is the measure of $\angle PSQ$?
Question : It is given that ABC $\cong$ PQR, AB = 5 cm, $\angle$B = $40^{\circ}$, and $\angle$A = $80^{\circ}$. Which of the following options is true?
Question : $\angle A, \angle B$ and $\angle C$ are three angles of a triangle. If $\angle A- \angle B=15^{\circ}, \angle B - \angle C=30^{\circ}$, then $\angle A, \angle B$ and $\angle C$ are:
Question : I is the incenter of a triangle ABC. If $\angle$ ABC = 65$^\circ$ and $\angle$ ACB = 55$^\circ$, then the value of $\angle$ BIC is:
Question : PQRS is a cyclic quadrilateral. If $\angle$P is three times of $\angle$R and $\angle$S is four times of $\angle$Q, then the sum of $\angle$S + $\angle$R will be:
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