Question

Points A, C, B and D are concyclic, AB is diameter and ∠AВС = 60°.

Assertion (A): ∠BAC = 60°.

Reason (R): AB is diameter so ∠ACB = 90° and ∠ABC + ∠BAC = 90°

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

A is false, R is true.

Explanation:

Join AC.

Since AB is diameter, that means:

∠ACB = 90° (Angles in a semicircle is a right angle)

In △ ABC, using angle sum property,

⇒ ∠ABC + ∠ACB + ∠BAC = 180°    ....(1)

⇒ 60° + 90° + ∠BAC = 180°

⇒ 150° + ∠BAC = 180°

⇒ ∠BAC = 180° - 150°

⇒ ∠BAC = 30°

So, assertion (A) is false.

From equation (1),

⇒ ∠ABC + 90° + ∠BAC = 180°

⇒ ∠ABC + ∠BAC = 180° - 90°

⇒ ∠ABC + ∠BAC = 90°

So, reason (R) is true.