ABCD is a cyclic quadrilateral as shown in figure. Find the angles of the cyclic quadrilateral.

Solution :

We know that the sum of opposite angles of cyclic quadrilateral is 180 degrees.cyclic quadrialteral

Angles A and C, Angles B and D form pairs of opposite angles in the given cyclic quadrilateral ABCD.

\(\angle\)A + \(\angle\)C = 180   and  \(\angle\)B + \(\angle\)D = 180

\(\implies\)  (4y + 20) + 4x = 180   and   (3y – 5) + (7x + 5) = 180

\(\implies\)  4x + 4y – 160 = 0     \(\implies\) x + y – 40 = 0      …………(1)

and  7x + 3y – 180 = 0         …………(2)

Multiplying equation (1) by and subtracting from equation (2), we get

4x – 60 = 0      \(\implies\)    x = 15

Put the value of x = 15 in equation (1), we get

15 + y – 40 = 0     \(\implies\)   y = 25

Hence, \(\angle\)A = 4y + 20 = 120 degrees

\(\angle\)B = 3y – 5 = 70 degrees

\(\angle\)C = 4x = 60 degrees

\(\angle\)D = 7x + 5 = 110 degrees

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