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To prove: x = 3y

Proof:

∠BOC = ∠BCO = y (Since. BC=BO)

Now, In Triangle OBC,

∠ ABO = ∠BOC + ∠BCO (Exterior angle of a triangle is equal to the interior opposite angles)

Therefore,

∠ABO = 2y

Again,

∠OBA = ∠ OAB = 2y

[Since AO = BO (since they are radii of same circle)]

In Triangle AOB,

∠ABO + ∠BAO + ∠BOA = 180 ( Angle sum property of triangle)

2y + 2y + ∠BOA = 180

∠BOA = 180 - 4y

Now,

∠AOC + x = 180 (linear pair)

180-4y + y + x = 180

180 - 3y + x = 180

x = 180-180 + 3y

Therefore, x = 3y