# 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.

(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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