The sum of any pair of opposite angles of a cyclic quadrilateral is:

Question

Accepted Answer

The correct option is 3:
To solve this problem, we need to recall a fundamental property of cyclic quadrilaterals. A cyclic quadrilateral is a quadrilateral whose all four vertices lie on the circumference of a circle.

The key theorem related to cyclic quadrilaterals states:
"The sum of any pair of opposite angles of a cyclic quadrilateral is $180^{\circ}$." These angles are also known as supplementary angles.

Let ABCD be a cyclic quadrilateral. According to this theorem:
$$\angle A + \angle C = 180^{\circ}$$
And
$$\angle B + \angle D = 180^{\circ}$$

Therefore, the sum of any pair of opposite angles of a cyclic quadrilateral is $180^{\circ}$.

The incorrect option is 1: A sum of $90^{\circ}$ (complementary angles) is not a property of opposite angles in a cyclic quadrilateral. This might be a misconception with right-angled triangles or other geometric figures.

The incorrect option is 2: A sum of $120^{\circ}$ is not a property of opposite angles in a cyclic quadrilateral. There is no general theorem that states this for cyclic quadrilaterals.

The incorrect option is 4: A sum of $360^{\circ}$ is the sum of all interior angles of any quadrilateral (cyclic or not), not just a pair of opposite angles. This option represents a common confusion between the sum of opposite angles and the sum of all angles in a quadrilateral.