O is the center of the circle, OB = BC and ∠BOC = 20°.

Statement (1) : x = 2 x 20° = 40°

Statement (2) : ∠BOC = 20°.

x = ∠OAB + 20° = ∠OBA + 20° = 40° + 20° = 60°

1. Both statements are true.

2. Both statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Given,

⇒ OB = OC

⇒ ∠BOC = ∠BCO = 20° (Angles opposite to equal sides of a triangle are always equal)

In △ OBC, using angle sum property,

⇒ ∠OBC + ∠BCO + ∠BOC = 180°

⇒ ∠OBC + 20° + 20° = 180°

⇒ ∠OBC + 40° = 180°

⇒ ∠OBC = 180° - 40°

⇒ ∠OBC = 140°

∠OBC and ∠OBA forms linear pairs of angle.

⇒ ∠OBC + ∠OBA = 180°

⇒ 140° + ∠OBA = 180°

⇒ ∠OBA = 180° - 140°

⇒ ∠OBA = 40°

Since OB = OA (Radii of same circle)

⇒ ∠OBA = ∠OAB = 40° (Angles opposite to equal sides of a triangle are always equal)

Using exterior angle property, the exterior angle of a triangle is equal to the sum of the two opposite interior angles.

In triangle OAC,

⇒ ∠EOA = ∠OAC + ∠OCA

⇒ x = ∠OAB + 20°

⇒ x = ∠OBA + 20° = 40° + 20° = 60°

So, statement 1 is false and statement 2 is true.

Hence, option 4 is the correct option.