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The diagram below a circle, centre O. PQ is a tangent to the circle at Q and PTOR is a straight line. QRST is a cyclic quadrilateral in which angle RTS = 350 and RT and QS are diameters. Giving reasons for your answer, find the size of:
Mathf4et121p1qn24

  1. Acute angle ROS. 
  2. Angle RQS. 
  3. Angle PQR.
  4. Angle QPT. 
  5. Angle PQT. 

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  1. Acute angle ROS. 
    Acute ∠ROS = 2∠RTS
    (∠ at centre)
    2 x 35 = 70º
  2. Angle RQS. 
    ∠RQS = ∠RTS
    ∠s in the same segment
    = 35º
  3. Angle PQR. 
    ∠PQS = 90º (∠ made by a tangent and radius)
    ∠PQR = ∠PQS + ∠RQS
    = 90º + 35º
    = 125º
  4. Angle QPT. 
    ∠QOT = ∠ROS = 35º
    (vertically opposite angles)
    In ΔPOQ, ∠QPT = 180o – (90 – 70)º
    = 20º (Sum of ∠s in a Δ)
  5. Angle PQT. 
    In ΔPQR, ∠PRQ = 180o – (125 + 20)º
    = 35º (∠ sum of Δ)
    ∠PQT = ∠PRQ (∠s in alternate segment)
    = 35º
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