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 = 180o



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.


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