In cyclic quadrilateral ABCD, ∠DAC = 27°; ∠DBA = 50° and ∠ADB = 33°. Calculate :

(i) ∠DBC,

(ii) ∠DCB,

(iii) ∠CAB.

(i) We know that,

Angles in same segment are equal.

∠DBC = ∠DAC = 27°.

Hence, ∠DBC = 27°.

(ii) We know that,

Angles in same segment are equal.

∠ACB = ∠ADB = 33°.

and,

∠ACD = ∠ABD = 50°.

From figure,

⇒ ∠DCB = ∠ACD + ∠ACB = 50° + 33° = 83°.

Hence, ∠DCB = 83°.

(iii) In quad. ABCD,

⇒ ∠DAB + ∠DCB = 180° [As sum of opposite angles in a cyclic quadrilateral = 180°]

⇒ ∠DAC + ∠CAB + ∠DCB = 180°

⇒ 27° + 83° + ∠CAB = 180°

⇒ ∠CAB + 110° = 180°

⇒ ∠CAB = 180° - 110° = 70°.

Hence, ∠CAB = 70°.