Question

A circle with centre at point O and ∠AOC = 160°.

Statement (1): Angle x = 100° and angle y = 80°.

Statement (2): The angle which, an arc of a circle subtends at the centre of the circle is double the angle which it subtends at any point on the remaining part of the circumference.

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

Statement 1 is false, and statement 2 is true.

Explanation:

We know that, the angle, subtended by an arc of a circle is double the angle subtended by it at any point on the remaining part of the circle.

So, statement 2 is true.

⇒ ∠AOC = 2x

⇒ 160° = 2x

⇒ x = `(160°)/2​` = 80°

We know that, sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

⇒ x + y = 180°

⇒ 80° + y = 180°

⇒ y = 180° − 80° = 100°

So, statement 1 is false.