In the given figure, ABCD is a cyclic quadrilateral in which ∠CAD = 25°, ∠ADB = 35° and ∠ABD = 50°. Calculate:

(i) ∠CBD

(ii) ∠CAB

(iii) ∠ACB

(i) We know that,

Angles in same segment are equal.

∠CBD = ∠CAD = 25°

Hence, ∠CBD = 25°.

(ii) We know that,

Angles in same segment are equal.Therefore,

⇒ ∠ACD = ∠ABD = 50°

⇒ ∠ACB = ∠ADB = 35°

From figure

⇒ ∠BCD = ∠ACD + ∠ACB

⇒ ∠BCD = 35° + 50°

⇒ ∠BCD = 85°.

We know that,

Sum of opposite angles in a cyclic quadrilateral = 180°.

⇒ ∠DAB + ∠DCB = 180°

⇒ ∠DAB = 180° - 85°

⇒ ∠DAB = 95°.

From figure,

⇒ ∠CAB = ∠DAB - ∠DAC

= 95° - 25°

= 70°.

Hence, ∠CAB = 70°.

(iii) Angles in same segment are equal. Therefore,

∠ACB = ∠ADB = 35°

Hence, ∠ACB = 35°.