A quadrilateral can be inscribed in a circle if and only if the opposite angles are

To determine if a quadrilateral can be inscribed in a circle, we use the property related to its opposite angles.

Detailed Explanation:

1. Inscribed Quadrilateral: A quadrilateral can be inscribed in a circle if and only if the sum of each pair of opposite angles equals $180^\circ$. This property stems from the nature of cyclic quadrilaterals.

2. Opposite Angles: If we denote the angles of the quadrilateral as $A$, $B$, $C$, and $D$, the conditions can be summarized as follows:

   • $A + C = 180^\circ$ (Sum of one pair of opposite angles)
   • $B + D = 180^\circ$ (Sum of the other pair of opposite angles)

Conclusion:

Therefore, the final statement is: A quadrilateral can be inscribed in a circle if and only if the opposite angles are supplementary, meaning each pair of opposite angles sum to $180^\circ$.