The lines `a_(1)x + b_(1)y + c_(1) = 0` and `a_(2)x + b_(2)y + c_(2) = 0` are perpendicular to each other , then `"_______"`.

Answer

To determine the condition for the lines \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) to be perpendicular, we can follow these steps:

### Step 1: Understand the concept of slopes
The slopes of two lines are crucial in determining whether they are perpendicular. If two lines are perpendicular, the product of their slopes is -1.

### Step 2: Convert the line equations to slope-intercept form
We need to rewrite both line equations in the form \( y = mx + c \) to identify their slopes.

1. For the first line \( a_1x + b_1y + c_1 = 0 \):
\[ b_1y = -a_1x - c_1 \]
\[ y = -\frac{a_1}{b_1}x - \frac{c_1}{b_1} \]
Thus, the slope \( m_1 = -\frac{a_1}{b_1} \).

2. For the second line \( a_2x + b_2y + c_2 = 0 \):
\[ b_2y = -a_2x - c_2 \]
\[ y = -\frac{a_2}{b_2}x - \frac{c_2}{b_2} \]
Thus, the slope \( m_2 = -\frac{a_2}{b_2} \).

### Step 3: Set up the condition for perpendicularity
According to the condition for perpendicular lines:
\[ m_1 \cdot m_2 = -1 \]
Substituting the slopes we found:
\[ \left(-\frac{a_1}{b_1}\right) \cdot \left(-\frac{a_2}{b_2}\right) = -1 \]
This simplifies to:
\[ \frac{a_1 a_2}{b_1 b_2} = -1 \]

### Step 4: Rearranging the equation
Multiplying both sides by \( b_1 b_2 \) gives:
\[ a_1 a_2 = -b_1 b_2 \]
Rearranging this leads to:
\[ a_1 a_2 + b_1 b_2 = 0 \]

### Final Answer
Thus, the condition for the lines \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) to be perpendicular is:
\[ a_1 a_2 + b_1 b_2 = 0 \]