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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