Question

A quadrilateral is inscribed in a circle. Two consecutive angles measure 88° and 108°. The angle opposite the 108° angle is labeled x°
. Find the value of x.

right answers only

Answer 1

In a cyclic quadrilateral, the sum of the opposite angles is \(180^\circ\). Given the two angles, let’s denote the angles as follows:

- Angle A = \(88^\circ\)
- Angle B = \(108^\circ\)
- Angle C = \(x^\circ\) (opposite angle to \(B\))
- Angle D = angle opposite to angle A (not needed explicitly)

Using the property of cyclic quadrilaterals:

1. Angle A + Angle C = 180°
\[
88^\circ + x^\circ = 180^\circ
\]
Therefore,
\[
x = 180^\circ - 88^\circ = 92^\circ
\]

2. Angle B + Angle D = 180°
\[
108^\circ + \text{Angle D} = 180^\circ
\]
This gives us:
\[
\text{Angle D} = 180^\circ - 108^\circ = 72^\circ
\]

We were only asked to find \(x\), which is:

\[
\boxed{92}
\]