Question

In the given figure, line PR touches the circle at point Q. Answer the following questions with the help of the figure.
(1) What is the sum of ∠ TAQ and ∠ TSQ ?
(2) Find the angles which are congruent to ∠ AQP. 
(3) Which angles are congruent to ∠ QTS ?
(4) ∠TAS = 65°, find the measure of ∠TQS and arc TS.
(5) If ∠AQP = 42°and ∠SQR = 58° find measure of ∠ATS.

Answer 1

(1)
Quadrilateral ATSQ is a cyclic quadrilateral.
∴ ∠TAQ + ∠TSQ = 180º            (Opposite angles of a cyclic quadrilateral are supplementary)
(2)
The angle between a tangent of a circle and a chord drawn from the point of contact is congruent to the angle inscribed in the arc opposite to the arc intercepted by that angle.
Here, PR is the tangent and AQ is the chord.
∴ ∠AQP ≅ ∠ATQ and ∠AQP ≅ ∠ASQ
⇒ ∠AQP ≅ ∠ATQ ≅ ∠ASQ
(3)
Angles inscribed in the same arc are congruent.
∴ ∠QTS ≅ ∠SAQ
Also, the angle between a tangent of a circle and a chord drawn from the point of contact is congruent to the angle inscribed in the arc opposite to the arc intercepted by that angle.
Here, PR is the tangent and QS is the chord.
∴ ∠QTS ≅ ∠SQR    
⇒ ∠QTS ≅ ∠SAQ ≅ ∠SQR
(4)
∠TQS = ∠TAS = 65º              (Angles inscribed in the same arc are congruent)
The measure of an inscribed angle is half of the measure of the arc intercepted by it.
∴ ∠TAS = `1/2` m(arc TS)
⇒ m(arc TS) = 2∠TAS = 2 × 65º = 130º
Thus, the measure of ∠TQS is 65º and arc TS is 130º.
(5)
The angle between a tangent of a circle and a chord drawn from the point of contact is congruent to the angle inscribed in the arc opposite to the arc intercepted by that angle.
Here, PR is the tangent and AQ is the chord.
∴ ∠ATQ = ∠AQP = 42º     .
Also, PR is the tangent and QS is the chord.
∴ ∠QTS ≅ ∠SQR = 58º
∠ATQ = ∠ATQ + ∠QTS = 42º + 58º = 100º
Thus, the measure of ∠ATQ is 100º.