We know that, in a cyclic quadrilateral, the sum of two opposite angles is 180°
∴∠A + ∠C = 180° and ∠B + ∠D = 180°
⇒ 2x + 4 + 2y + 10 = 180 and y + 3 + 4x – 5 = 180
⇒ 2x + 2y = 180 – 14 and 4x + y – 2 = 180
⇒ x + y = 83 and 4x + y = 182
So, we get pair of linear equation i.e.
x + y = 83 …(i)
4x + y = 182 …(ii)
On subtracting Eq.(i) from (ii), we get
4x + y – x – y = 182 – 83
⇒ 3x = 99
⇒ x = 33
On putting the value of x = 33 in Eq. (i) we get,
33 + y = 83
⇒ y = 83 – 33 = 50
On putting the values of x and y, we calculate the angles as
∠A = (2x + 43)° = 2(33) + 4 = 66 + 4 = 70°
∠B = (y + 3)° = 50 + 3 = 53°
∠C = (2y + 10)° = 2(50) + 10 = 100 + 10 = 110°
and ∠D = (4x - 5)° = 4(33) – 5 = 132 – 5 = 127°
Hence, the angles are ∠A = 63°, ∠B = 57°, ∠C = 117°, ∠D = 123°