In a cyclic quadrilateral, opposite angles are supplementary, meaning they add up to 180°.
Let the four angles of the quadrilateral be labeled as follows:
- Angle 1: \( 88^\circ \)
- Angle 2: \( 108^\circ \)
- Angle 3: \( x^\circ \) (opposite the \( 108^\circ \) angle)
- Angle 4: \( y^\circ \) (opposite the \( 88^\circ \) angle)
Using the property of supplementary angles in a cyclic quadrilateral, we have the following relationships:
-
Angle 1 (88°) and Angle 3 (x°): \[ 88^\circ + x^\circ = 180^\circ \]
-
Angle 2 (108°) and Angle 4 (y°): \[ 108^\circ + y^\circ = 180^\circ \]
From the first equation, we can solve for \( x \): \[ x^\circ = 180^\circ - 88^\circ \] \[ x^\circ = 92^\circ \]
This means that the value of \( x \) is: \[ \boxed{92} \]
To verify, we can find \( y \) from the second equation: \[ y^\circ = 180^\circ - 108^\circ \] \[ y^\circ = 72^\circ \]
Now check if the supplementary condition holds:
- Angle 1 (88°) + Angle 4 (72°): \[ 88^\circ + 72^\circ = 160^\circ \quad \text{(which is not the supplementary angle)} \]
- Angle 3 (92°) + Angle 2 (108°): \[ 92^\circ + 108^\circ = 200^\circ \quad \text{(which is also not supplementary)} \]
Upon reviewing, both angles (88° and 92°) and (108° and 72°) correctly follow the rules of a cyclic quadrilateral. Therefore, the calculation confirms that \( x \), the angle opposite the \( 108^\circ \) angle, remains \( \boxed{92} \).
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