Which of the following options is correct :-
A pair of adjacent angles have.
(i)Common vertex
(ii)Common Arm.
(iii)Non Common arms are an opposite sides of common arms.
(iv)Non Common arms are on the same side of common arms.

Options

A. (i) & (ii) are sufficient
B. (i),(ii) & (iii) are sufficient
C. (i) ,(ii) & (iv) are sufficient
D. All are sufficient

Answer

To determine which of the given options is correct regarding a pair of adjacent angles, let's analyze each option step by step.

### Step 1: Understand the Definition of Adjacent Angles
Adjacent angles are two angles that share a common vertex and a common arm (side), while their non-common arms are on opposite sides of the common arm.

### Step 2: Analyze Each Option
1. **Option (i): Common vertex**
- **Explanation**: For two angles to be adjacent, they must share a common vertex.
- **Conclusion**: This option is correct.

2. **Option (ii): Common Arm**
- **Explanation**: Adjacent angles must have one arm in common.
- **Conclusion**: This option is correct.

3. **Option (iii): Non-common arms are on opposite sides of common arms**
- **Explanation**: The non-common arms of adjacent angles must be on opposite sides of the common arm.
- **Conclusion**: This option is correct.

4. **Option (iv): Non-common arms are on the same side of common arms**
- **Explanation**: This statement contradicts the definition of adjacent angles, as the non-common arms must be on opposite sides of the common arm.
- **Conclusion**: This option is incorrect.

### Final Conclusion
The correct options are (i), (ii), and (iii). Option (iv) is incorrect.

### Summary of Correct Options
- (i) Common vertex: Correct
- (ii) Common arm: Correct
- (iii) Non-common arms are on opposite sides of common arms: Correct
- (iv) Non-common arms are on the same side of common arms: Incorrect