ABCD is a cyclic quadrilateral (see Fig. 3.7). Find the angles of the cyclic quadrilateral.

Question

Philoid · Accepted Answer

We know that the sum of the measures of opposite angles in a cyclic quadrilateral is 180°. Therefore, ∠ A + ∠ C = 180

4y + 20 - 4x = 180 - 4x + 4y = 160

x - y = - 40 (i)

Also, ∠ B + ∠ D = 180 3y - 5 - 7x + 5 = 180 - 7x

+ 3y = 180 (ii)

Multiplying equation (i) by 3, we obtain 3x - 3y = - 120 (iii)

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

- 7x + 3x = 180 - 120

- 4x = 60

x = -15

By using equation (i), we obtain x - y = - 40

-15 - y = - 40

y = -15 + 40 = 25

∠ A = 4y + 20 = 4(25) + 20 = 120°

∠ B = 3y - 5 = 3(25) - 5 = 70°

∠ C = - 4x = - 4(- 15) = 60°