Mathematics
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 to be 90°.
Reason (R) : In △ADB,
∠ADB = 180° - 95° - 40° = 45°
∴ Angle ADC = 45° + 45° = 90°
(i) A is true, R is false
(ii) A is true, R is true
(iii) A is false, R is false
(iv) A is false, R is true
Answer
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.
Hence, Option 2 is the correct option.
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