In figure 3.91, line PR touches the circle at point Q. (1) Sum of ∠TAQ and ∠TSQ? (2) Angles congruent to ∠AQP? (3) Angles congruent to ∠QTS? (4) ∠TAS = 65°, find ∠TQS and arc TS. (5) If ∠AQP = 42° and ∠SQR = 58° find ∠ATS.

Question

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 . Figure: Points T, S, Q, A lie on the circle in that order (T at the top, S on the right, Q at the bottom, A on the left). Line PR touches the circle at Q, with P on the side of A and R on the side of S. Segments TA, TS, TQ, AS, AQ and QS are drawn.

Doubtly AI · Answer

Answer: (1) 180^° (2) ∠ ASQ , ∠ ATQ (3) ∠ SQR , ∠ SAQ (4) ∠ TQS = 65^° , arc TS = 130^° (5) ∠ ATS = 100^° (1) □TAQS is cyclic and ∠ TAQ , ∠ TSQ are opposite angles, so their sum is 180^° . (2) ∠ AQP is the angle between tangent QP and chord QA, so it equals half of arc AQ. The inscribed angles intercepting arc AQ are ∠ ASQ and ∠ ATQ . So ∠ AQP cong ∠ ASQ cong ∠ ATQ . (3) ∠ QTS intercepts arc QS. So do the tangent–chord angle ∠ SQR and the inscribed angle ∠ SAQ . So ∠ QTS cong ∠ SQR cong ∠ SAQ . (4) ∠ TQS and ∠ TAS are inscribed in the same arc, so ∠ TQS = 65^° , and m(arc TS) = 2 × 65^° = 130^° (5) m(arc AQ) = 2 × 42° = 84° and m(arc QS) = 2 × 58° = 116°. ∠ ATS intercepts arc AQS: ∠ ATS = 1/2(84^° + 116^°) = 100^°

In figure 3.91, line PR touches the circle at point Q. (1) Sum of ∠TAQ and ∠TSQ? (2) Angles congruent to ∠AQP? (3) Angles congruent to ∠QTS? (4) ∠TAS = 65°, find ∠TQS and arc TS. (5) If ∠AQP = 42° and ∠SQR = 58° find ∠ATS.

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