Question

In 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 angle TAQ and
angle TSQ ?
(2) Find the angles which are
congruent to angle AQP
(3) Which angles are congruent to
angle QTS?
(4) ZTAS = 65°, find the measure of ZTQS and arc TS.
(5) If angle AQP = 42°and angle SQR = 58° find measure of angle ATS.​

Answer 1

Answer:

100°

Step-by-step explanation:

(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°