Question

O is centre of the circle and ∠AOC = 120°.

Statement (1): ∠ABC = 120°

Statement (2): ∠ABC + ∠ADC = 180°
⇒ ∠ABC + 60° = 180°

Options

• Both the statements are true.

• Both the statements are false.

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

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

Answer 1

Both the statements are true.

Explanation:

Since, the angle, which an arc of a circle subtends at the center of the circle is double the angle which it subtends at any point on the remaining part of the circumference.

⇒ ∠AOC = 2 × ∠ADC

⇒ 120° = 2 × ∠ADC

⇒ ∠ADC = `(120°)/2`​ = 60°

ABCD form a cyclic quadrilateral and sum of opposite angles of cyclic quadrilateral is 180°.

⇒ ∠ADC + ∠ABC = 180°

⇒ 60° + ∠ABC = 180°

⇒ ∠ABC = 180° − 60°

⇒ ∠ABC = 120°

So, both statement are true.