Q24 of 62 Page 10

 

AB is a chord of the circle C(O, r)      
∠AOC = 2∠ABC    ....... (i)
∠BOC = 2∠BAC   ....... (ii)     (∵ Angles at the centre formed by an arc is twice the angle made by the same arc 
                                            in the alternate segment)
Adding equations (i) and (ii), we get
∠AOC + ∠BOC = 2∠ABC + 2∠BAC 
⇒ ∠AOB = 2(∠CAB + ∠CBA)

More from this chapter

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22 In the Fig, O is the centre of the circle, prove that,  ∠a = ∠b + ∠c.
  23 In the figure, P is a point on the chord BC such that AB = AP. Prove that, CP = CQ.
  25 AB is a diameter of a circle C(O, r) and radius OD ⊥ AB. If C is any point on arc DB, then find ∠BAD and ∠CAD.
If ∠BAC = 20°, find ∠DAC.  26   AB and AC are two chords of a circle and the diameter passing through 'A' bisects the angle ∠BAC. Prove that AB = AC.