Proof that the sum of opposite angles in a cyclic quadrilateral is 180 degrees:

To prove that the sum of either pair of opposite angles in a cyclic quadrilateral is 180 degrees, we can use the properties of angles in a circle and the fact that the sum of angles in a triangle is 180 degrees.

Given:
A cyclic quadrilateral ABCD, where the opposite angles are ∠A and ∠C, and ∠B and ∠D.

Proof:

Step 1: Draw the diagonals AC and BD of the cyclic quadrilateral ABCD.

Step 2: Identify the triangles formed by the diagonals and sides of the quadrilateral: triangle ABC and triangle CDA.

Step 3: Apply the property of angles in a circle: the angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circumference.

Step 4: Since ABCD is a cyclic quadrilateral, the opposite angles A and C are subtended by the same arc, and similarly, the opposite angles B and D are subtended by the same arc.

Step 5: Considering triangle ABC, the sum of angles A and C is equal to the angle subtended by arc BC at any point on the circumference.

Step 6: Similarly, considering triangle CDA, the sum of angles C and A is equal to the angle subtended by arc AD at any point on the circumference.

Step 7: Since angles subtended by the same arc at any point on the circumference are equal, we can conclude that the sum of angles A and C is equal to the sum of angles C and A.

Step 8: Applying the property of the sum of angles in a triangle, we know that the sum of angles in triangle ABC is 180 degrees.

Step 9: Therefore, the sum of angles A and C is equal to 180 degrees.

Conclusion:
Thus, we have proved that the sum of either pair of opposite angles in a cyclic quadrilateral is 180 degrees.