The angles of a cyclic quadrilateral ABCD are ∠A = (6x + 10)°, ∠B = (5x)°, ∠C = (x + y)° and ∠D = (3y – 10)°.

Find x and y and hence the values of the four angles.

We know that, by property of cyclic quadrilateral,

Sum of opposite angles = 180^o

Since
and

So, 7x + y = 170 …(i)

and

Since
and

So, 5x + 3y = 190 …(ii)

On multiplying Eq. (i) by 3 and then subtracting, we get

3(7x + y) – (5x + 3y) = 3(170) – 190

16x = 320

x = 20

On putting x = 20 in Eq. (i), we get

7(20) + y = 170

So, y = 30

And hence

= 20 + 30 = 50

Hence, the required values of x and y are 20 and 30 respectively and the values of the four angles i.e., and, are 130, 100, 50, and 80, respectively.