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# ABCD is a cyclic quadrilateral (see figure). Find the angles of the cyclic quadrilateral.

We know that the sum of the measures of opposite angles in a cyclic quadrilateral is :

$$\angle$$A + $$\angle$$C = 180°
=>$$4y + 20 - 4x = 180°$$
=>$$-4x + 4y = 160°$$
=>$$x - y = -40°$$ ………(1)
Also $$\angle$$B + $$\angle$$D = 180°
=>$$3y - 5 - 7x + 5 = 180°$$
=>$$-7x + 3y = 180°$$………(2)

Multiplying equation (1) by 3:
$$3x - 3y = - 120°$$ ……….(3)

$$-4x = 60° => x = -15°$$
$$-15 - y = -40°$$
=>$$y = -15 + 40 =25$$
$$\angle$$A = $$4y + 20 = 4(25) + 20 = 120°$$
$$\angle$$B = $$3y - 5 = 3(25) - 5 = 70°$$
$$\angle$$C = $$-4x = -4(-15) = 60°$$
$$\angle$$D = $$-7x + 5 = -7(-15) + 5 = 110°$$