Question

find the value of each variable.
a. ( x = 75^circ ) and ( y = 78^circ )
b. ( x = 78^circ ) and ( y = 65^circ )
c. ( x = 102^circ ) and ( y = 115^circ )
d. ( x = 71.5^circ ) and ( y = 71.5^circ )

Explanation:

Step1: Recall cyclic quadrilateral property

In a cyclic quadrilateral, the sum of each pair of opposite angles is \(180^\circ\) (supplementary). So, if we have angles \(115^\circ\) and \(y^\circ\) as opposite, and \(102^\circ\) and \(x^\circ\) as opposite.

Step2: Calculate \(x\)

Using the property, \(x + 102=180\). Solving for \(x\), we get \(x = 180 - 102 = 78^\circ\).

Step3: Calculate \(y\)

Also, \(y + 115 = 180\). Solving for \(y\), we get \(y = 180 - 115 = 65^\circ\).

Answer:

B. \(x = 78^\circ\) and \(y = 65^\circ\)