To calculate angles in a cyclic quadrilateral, we can utilize the property that the opposite angles sum to $180^\circ$.

A cyclic quadrilateral is a four-sided figure in which all vertices lie on the circumference of a circle. One of its fundamental properties is that the sum of each pair of opposite angles is always $180^\circ$. This characteristic is referred to as the supplementary angles property. For instance, in a cyclic quadrilateral labeled as $ABCD$, the relationships are expressed as follows:

If you need to determine a specific angle, you can apply this property effectively. Suppose you know the measures of three angles in the quadrilateral. You can find the fourth angle by subtracting the measure of the known opposite angle from $180^\circ$. For example, if you are aware of angles $A$, $B$, and $C$, you can calculate angle $D$ using the formula:

Moreover, additional geometric properties and theorems pertaining to circles and triangles can assist in finding unknown angles. For instance, the exterior angle of a cyclic quadrilateral is equal to the measure of the interior opposite angle. This principle can be particularly useful in more complex problems that require multiple steps to ascertain the unknown angles.

Finally, it is crucial to verify that the quadrilateral in question is indeed cyclic before applying these properties, as they do not apply to non-cyclic quadrilaterals.