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



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°


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