Question

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:

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

Answer 1

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