ABCD is a cyclic quadrilateral finds the angles of the cyclic quadrilateral.

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#### Solution

We know that the sum of the measures of opposite angles in a cyclic quadrilateral is 180°.

Therefore, ∠A + ∠C = 180

4*y* + 20 − 4*x* = 180

− 4*x* + 4*y* = 160

*x* − *y* = − 40 (*i*)

Also, ∠B + ∠D = 180

3*y* − 5 − 7*x* + 5 = 180

− 7*x* + 3*y* = 180 (*ii*)

Multiplying equation (*i*) by 3, we obtain

3*x* − 3*y* = − 120 (*iii*)

Adding equations (*ii*) and (*iii*), we obtain

− 7*x* + 3*x* = 180 − 120

− 4*x* = 60

*x* = −15

By using equation (*i*), we obtain

*x *− *y* = − 40

−15 − *y* = − 40

*y* = −15 + 40 = 25

∠A = 4*y* + 20 = 4(25) + 20 = 120°

∠B = 3*y* − 5 = 3(25) − 5 = 70°

∠C = − 4*x* = − 4(− 15) = 60°

∠D = − 7*x* + 5 = − 7(−15) + 5 = 110°

Concept: Equations Reducible to a Pair of Linear Equations in Two Variables

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