If `a + ib = c + id`, then

Options

A. `a^(2) + c^(2) = 0`
B. `b^(2)+ c^(2) = 0`
C. `b^(2) + d^(2) = 0`
D. `a^(2) + b^(2) = c^(2) + d^(2)`

Explanation

To solve the equation \( a + ib = c + id \), we start by equating the real and imaginary parts.

### Step 1: Equate the real parts
From the equation \( a + ib = c + id \), we can separate the real and imaginary parts:
- Real part: \( a = c \)
- Imaginary part: \( b = d \)

### Step 2: Establish relationships between the squares
From the results of Step 1, we can write:
- \( a^2 = c^2 \)
- \( b^2 = d^2 \)

### Step 3: Formulate the difference of squares
Using the identities from Step 2:
- \( a^2 - c^2 = 0 \)
- \( b^2 - d^2 = 0 \)

This tells us that both \( a^2 \) and \( c^2 \) are equal, and both \( b^2 \) and \( d^2 \) are equal.

### Step 4: Consider the magnitudes
We can also consider the magnitudes of the complex numbers:
\[ |a + ib| = |c + id| \]
This translates to:
\[ \sqrt{a^2 + b^2} = \sqrt{c^2 + d^2} \]

### Step 5: Square both sides
Squaring both sides gives us:
\[ a^2 + b^2 = c^2 + d^2 \]

### Conclusion
The correct conclusion from the above steps is that:
\[ a^2 + b^2 = c^2 + d^2 \]
This corresponds to option D from the given choices.

### Final Answer
The correct option is D: \( a^2 + b^2 = c^2 + d^2 \).

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