In the figure, given below, find :

(i) ∠BCD,

(ii) ∠ADC,

(iii) ∠ABC.

Show steps of your working.

(i) We know that,

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

⇒ ∠BCD + ∠DAB = 180°

⇒ ∠BCD + 105° = 180°

⇒ ∠BCD = 180° - 105° = 75°.

Hence, ∠BCD = 75°.

(ii) Sum of co-interior angles in a trapezium = 180°.

⇒ ∠ADC + ∠DAB = 180°

⇒ ∠ADC + 105° = 180°

⇒ ∠ADC = 180° - 105° = 75°.

Hence, ∠ADC = 75°.

(iii) We know that, the sum of angles in a quadrilateral is 360°

So,

⇒ ∠ADC + ∠DAB + ∠BCD + ∠ABC = 360°

⇒ 75° + 105° + 75° + ∠ABC = 360°

⇒ ∠ABC = 360° - 255°

⇒ ∠ABC = 105°.

Hence, ∠ABC = 105°.