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

Answer 1

To find the value of \( x \), we can use the property of cyclic quadrilaterals (quadrilaterals inscribed in a circle). The property states that opposite angles of a cyclic quadrilateral are supplementary, which means that the sum of the measures of each pair of opposite angles is \( 180^\circ \).

In this case, we have the following angles:

- Angle \( A = 88^\circ \)
- Angle \( B = 108^\circ \) (adjacent to \( A \))
- Angle \( C = x^\circ \) (opposite \( B \))
- Angle \( D \) (opposite \( A \))

From the property of the cyclic quadrilateral, we have:

\[
A + C = 180^\circ
\]

Substituting the value of angle \( A \):

\[
88^\circ + x^\circ = 180^\circ
\]

Now, we can solve for \( x \):

\[
x^\circ = 180^\circ - 88^\circ
\]
\[
x^\circ = 92^\circ
\]

So the value of \( x \) is \( \boxed{92} \).