By the theorem of parallel axes:

Options

A. `I=I_(G)-md^(2)`
B. `I_(G)=I+md^(2)`
C. `I+I_(G)=md^(2)`
D. `I_(G)=I-md^(2)`

Answer

To solve the problem using the theorem of parallel axes, we need to understand the theorem itself and then apply it to determine the correct relation among the given options.

### Step-by-Step Solution:

1. **Understanding the Parallel Axis Theorem**:
The parallel axis theorem states that the moment of inertia (I) of a body about any axis parallel to an axis through its center of mass (I_g) can be calculated using the formula:
\[ I = I_g + md^2 \]
where:
- \(I\) = moment of inertia about the new axis
- \(I_g\) = moment of inertia about the center of mass axis
- \(m\) = mass of the body
- \(d\) = distance between the two parallel axes

2. **Analyzing the Given Options**:
We need to evaluate the given options based on the parallel axis theorem:
- **Option 1**: \(I = I_g - md^2\) (Incorrect)
- **Option 2**: \(I = I_g + md^2\) (Correct)
- **Option 3**: \(I + I_g = md^2\) (Incorrect)
- **Option 4**: \(I_g = I - md^2\) (This can be rearranged to \(I = I_g + md^2\), which is correct)

3. **Identifying the Correct Relation**:
From our understanding of the parallel axis theorem, the correct relation is:
\[ I = I_g + md^2 \]
This corresponds to **Option 2**. However, since Option 4 can be rearranged to give the same equation, it can also be considered correct.

4. **Conclusion**:
The correct relations based on the parallel axis theorem are:
- **Option 2**: \(I = I_g + md^2\) (directly from the theorem)
- **Option 4**: \(I_g = I - md^2\) (rearranged form)

### Final Answer:
The correct options are **Option 2** and **Option 4**.