In a cyclic quadrilateral ABCD. ∠A= (2x+4)°, ∠B = (y+3)°, ∠C = (2y+10)°, and ∠D = (4x–5)°. Find the measure of each angle.

Cyclic Quadrilateral according to the question is shown above.

Now by property of cyclic quadrilateral, we know that opposite angles of a cyclic quadrilateral are supplementary.

Therefore,

Sum of Adjacent angles of cyclic quadrilateral = 180°

So, we have,

(2x + 4) + (2y + 10) = 180°

2x + 2y = 166°…..(i)

And also,

(y + 3) + (4x – 5) = 180°

4x +y = 182° ……..(ii)

Solving equations (i) and (ii)

Eq (i) – 2 × eq (ii)

2x + 2y – 8x – 2y = 166° – 364°

–6x = – 198°

x = 33°

Putting the value of x in eq (i) we get,

2x + 2y = 166°

2 × 33° + 2y = 166°

2y = 100°

y = 50°

Now each angle of the cyclic quadrilateral is,

∠A = (2x + 4)° = 2 × 33° + 4° = 70°

∠B = (y + 3)° = 50° + 3° = 53°

∠C = (2y + 10)° = 2 × 50° + 10° = 110°

∠D = (4x – 5)° = 4 × 33° – 5° = 127°