Q. 134.4( 19 Votes )

# In figure 3.91, 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) As TAQS is a cyclic quadrilateral,

∠TAQ + ∠TSQ = 180° (Sum of opposite angles of a cyclic quadrilateral is 180° )

(2) ∠ASQ and ∠ATQ

(3) ∠ QAS and ∠SQR

(4) ∠TAS = 65°

∠ TQS = ∠ TAS = 65° (angle by same arc TS in the same sector)

m(arc TS) = ∠TQS + ∠TAS

⇒ m(arc TS) = 65 + 65 = 130°

(5) ∠AQP + ∠AQS + ∠SQR = 180°

⇒ 42 + ∠AQS + 58 = 180

⇒ ∠AQS + 100 = 180

⇒ ∠AQS = 80

∠ AQS + ∠ ATS = 180° (opposite angles of a cyclic quadrilateral)

⇒ 80 + ∠ATS = 180

⇒ ∠ATS = 100°

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