$A B C D$ is a cyclic quadrilateral such that $∠ A = (4 y + 20), ∠ B = (3 y - 5), ∠ C = (- 4 x)$ and $∠ D = (7 x + 5)$ . Find the four angles.

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Solution

In quadrilateral ABCD,
$∠ A + ∠ C = 180 °$
$\Rightarrow 4 y + 20 - 4 x = 180 °$
$- 4 x + 4 y = 160 °$
$\Rightarrow - x + y = 40 °$
$\Rightarrow - 3 x + 3 y = 120 ° \to 1$
$∠ B + ∠ D = 180 °$
$3 y - 5 + 7 x + 5 = 180 °$
$7 x + 3 y = 180 ° \to 2$
Solving equation $1$ and $2$ , we get
$7 x + 3 y + 3 x - 3 y = 180 ° - 120 ° \Rightarrow 10 x = 60 ° \Rightarrow x = 6$
& $- 6 + y = 40 ° \Rightarrow y = 46 °$
$∴ ∠ A = 4 \times 46 ° + 20 = 204 °, ∠ B = 3 \times 46 - 5 = 133 °$
$∠ C = - 4 \times 6 = - 24 °, ∠ D = 7 \times 6 + 5 = 47 °$

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