Question

i) consider the following figure,

if pqrs is a quadrilateral inscribed in a circle, then the opposite angles of the quadrilateral are

ii) the values of x and y are degrees and degrees respectively.

Explanation:

Step1: Recall the property of cyclic quadrilaterals

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the sum of opposite angles is \(180^{\circ}\).

Step2: Find the value of \(x\)

Since \(x\) and \(82^{\circ}\) are opposite angles in the cyclic quadrilateral \(ABCD\) (assuming the given quadrilateral is \(ABCD\) as per the figure labels), we use the property \(x + 82^{\circ}=180^{\circ}\).

$$x=180^{\circ}- 82^{\circ}$$

$$x = 98^{\circ}$$

Step3: Find the value of \(y\)

Since \(y\) and \(68^{\circ}\) are opposite angles in the cyclic quadrilateral \(ABCD\), we use the property \(y+68^{\circ}=180^{\circ}\).

$$y = 180^{\circ}-68^{\circ}$$

$$y=112^{\circ}$$

Answer:

\(x = 98\) degrees and \(y = 112\) degrees.