Statement-I: If the sum of a pair of opposite angles of a quadrilateral is `180^(@)`, then the quadrilateral is cyclic
Statement-II : The angle subtended by an are at the centre is half the angle subtended by it at any point on the remaining part of the circle.

Options

A. Both Statement-I and Statement-II are true.
B. Both Statement-I and Statement-II are false
C. Statement-I is true but Statement-II is false.
D. Statement-I is false but Statement-II is true.

Answer

To solve the problem, we need to analyze the two statements provided and determine their truth values.

### Step 1: Analyze Statement I
**Statement I:** If the sum of a pair of opposite angles of a quadrilateral is \(180^\circ\), then the quadrilateral is cyclic.

1. **Draw a Quadrilateral**: Let's denote the quadrilateral as \(ABCD\).
2. **Assume Opposite Angles**: Assume that \( \angle A + \angle C = 180^\circ \) and \( \angle B + \angle D = 180^\circ \).
3. **Cyclic Quadrilateral Definition**: A quadrilateral is cyclic if all its vertices lie on a single circle.
4. **Use the Property of Cyclic Quadrilaterals**: In a cyclic quadrilateral, the sum of opposite angles is \(180^\circ\).
5. **Conclusion**: Since we have shown that if the sum of opposite angles is \(180^\circ\), then the quadrilateral can be inscribed in a circle, thus confirming that the statement is true.

### Step 2: Analyze Statement II
**Statement II:** The angle subtended by an arc at the center is half the angle subtended by it at any point on the remaining part of the circle.

1. **Draw a Circle**: Let \(O\) be the center of the circle and \(P\) and \(Q\) be points on the circumference.
2. **Identify Angles**: Let the angle subtended at the center \(O\) by arc \(PQ\) be \( \angle POQ \) and the angle subtended at point \(R\) on the circumference be \( \angle PRQ \).
3. **Use the Inscribed Angle Theorem**: According to this theorem, the angle subtended at the center (i.e., \( \angle POQ \)) is actually twice the angle subtended at any point on the circumference (i.e., \( \angle PRQ \)).
4. **Conclusion**: Therefore, the statement that the angle subtended at the center is half the angle subtended at any point on the circumference is false. It should be the other way around: the angle at the center is twice that at the circumference.

### Final Conclusion
- **Statement I is True**.
- **Statement II is False**.

Thus, the correct answer is that Statement I is true and Statement II is false.