Question

In the given circle, ∠BAD = 95°, ∠ABD = 40° and ∠BDC = 45°.

Assertion (A): To show that AC is a diameter, the angle ADC or angle ABC need to be proved equal to 90°.

Reason (R): In △ADB,
∠ADB = 180° − 95° − 40° = 45°
∴ ∠ADC = 45° + 45° = 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

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

Explanation:

We know that,

Angle in semicircle is a right angle.

If AC is the diameter, then ∠ADC = ∠ABC = 90°.

∴ Assertion (A) is true.

From figure,

In △ADB,

By angle sum property of triangle,

∴ ∠ADB + ∠DBA + ∠BAD = 180°

⇒ ∠ADB + 40° + 95° = 180°

⇒ ∠ADB + 135° = 180°

⇒ ∠ADB = 180° − 135° = 45°

From figure,

⇒ ∠ADC = ∠ADB + ∠BDC = 45° + 45° = 90°

∴ Reason (R) is true.