SSC CGL Model Paper 18

∠ROS = 40°
RO = OS = OP = OQ = radius
∴ ∠OSQ = ∠SQO = y
∠PRO = ∠OPR = x
In ΔORS,
∠RSO = ∠SRO = 70°
∠RSQ + ∠RPQ = 180°
(The sum of opposite angles of a cyclic quadrilateral is 180°)
∠RSO + ∠OSQ + ∠RPO = 180°
70 + y + x =180°
x + y = 110° ...(i)
∠PRT = 180°
⇒ ∠PRO + ∠ORS + ∠TRS =180°
⇒ x + 70 °+∠TRS = 180°
∴ ∠TRS = 110°- x ...(ii)
Similarly,
∠TSR =110° - y ...(iii)
Adding Eqs. (ii) and (iii),
∠TRS + ∠TSR = 110° - x + 110° - y
⇒ ∠TRS + ∠TSR = 220° - (x + y)
= 220° - 110° [from Eq. (i)]
∠TRS + ∠TSR = 110°
⇒ ∠RTS =180°-110° = 70°
⇒ ∠RTQ = ∠RTS = ∠PTQ =70°