Question

ABCD is a cyclic quadrilateral, BD and AC are its diameters. Also, ∠DBC = 50°.

Assertion (A): ∠BAC = 40°.

Reason (R): ∠BAC = ∠BDC = 180° − (50° + 90°) = 40°

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

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

Explanation: 

Since BD and AC are diameters, that means:

∠ABC and ∠BCD are right angles (Angles in a semicircle is a right angle)

In △ DBC, using angle sum property,

⇒ ∠DBC + ∠BCD + ∠BDC = 180°

⇒ 50° + 90° + ∠BDC = 180°

⇒ 140° + ∠BDC = 180°

⇒ ∠BDC = 180° − 140°

⇒ ∠BDC = 40°

We know that, angles in the same segment of a circle are equal.

⇒ ∠BAC = ∠BDC

⇒ ∠BAC = 40°

So, assertion and reason are true and reason clearly explains assertion.