Opposite angles in a cyclic quadrilateral

Cyclic Quadrilateral:
- A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle.

Sum of opposite angles

In a cyclic quadrilateral, the sum of the opposite angles is always equal to 180 degrees. This property is known as the opposite angles of a cyclic quadrilateral theorem.

Explanation:
- Let's consider a cyclic quadrilateral ABCD where the vertices A, B, C, and D lie on the circumference of a circle.
- The opposite angles in this quadrilateral are angle A and angle C, and angle B and angle D.
- According to the opposite angles of a cyclic quadrilateral theorem, angle A + angle C = 180 degrees and angle B + angle D = 180 degrees.
- This property holds true for all cyclic quadrilaterals, regardless of the size or shape of the quadrilateral.

Proof:
- The proof of this theorem involves using the properties of angles in a circle and the properties of cyclic quadrilaterals.
- By applying these properties and using the fact that the sum of angles in a quadrilateral is 360 degrees, we can prove that the sum of opposite angles in a cyclic quadrilateral is always 180 degrees.

In conclusion, the sum of the opposite angles in a cyclic quadrilateral is always equal to 180 degrees, which is a unique property of these special quadrilaterals.