In the given figure, O is the centre of the circle. PQ is a tangent to the circle at A. If ∠ PAB = 58°, find ∠ ABQ and ∠ AQB. 

Given: O is the centre of the circle. PQ is a tangent to the circle at A. ∠ PAB = 58°. 
To find: ∠ ABQ and ∠ AQB.
∠ BAR = 90° (Angle in a semicircle)
∠ ARB = ∠ PAB = 58° (Alternate segment theorem) 
∠ ABQ = 180° - (∠ BAR + ∠ ARB) (Angle sum property of a triangle)
         = 180° - (90° + 58°) 
         = 180° – 148° = 32° 
∠ QAR = ∠ ABR = 32° (Alternate segment theorem)
and ∠ AQB = 180° – (∠ ABQ + ∠ BAQ) (Angle sum property of a triangle)
               = 180° – (32° + 90° + 32°) 
               = 180° – 154° = 26°.