Statements

Reasons 
1) ∠ACB = ∠ADB 
1) Angles in the same segment. 
2) ∠BAC = ∠BDC 
2) Angles in the same segment 
3)∠ACB + ∠BAC = ∠ADB + ∠BDC 
3) Addition property 
4) ∠ACB + ∠BAC = ∠ADC 
4) Add ∠ABC on both sides. 
5) ∠ABC + ∠ACB + ∠BAC = ∠ABC + ∠ADC 
5) From Above.

6) 180^{o} = ∠ABC + ∠ADC

6) Sum of the angle of a triangle is 180^{o}

7) ∠B + ∠D = 180^{o}

7) Opposite angles of cyclic quadrilateral. 
8) ∠A + ∠B + ∠C + ∠D 
8) Measure of a quadrilateral. 
9) ∠A + ∠C = 360^{o}  (∠B + ∠D)

9) From Above. 
10) ∠A + ∠C = 360^{o}  180^{o} = 180^{o}

10) Angle sum property

11) ∠A + ∠C = 180^{o} and ∠B + ∠D = 180^{o}

11) From above .So opposite angles are supplementary. 