State True or False: Acute angle between two lines with slopes m and `m_(2)` is given by `tantheta=|(m_(2)-m_(1))/(1+m_(1)m_(2))|`

Answer

To determine whether the statement is true or false, we need to analyze the given formula for the acute angle between two lines with slopes \( m_1 \) and \( m_2 \).

### Step-by-Step Solution:

1. **Understanding the Slopes**:
The slopes of two lines are denoted as \( m_1 \) and \( m_2 \). These slopes represent the tangent of the angles that the lines make with the positive x-axis.

2. **Defining the Angles**:
Let \( \phi_1 \) be the angle of inclination of the first line with slope \( m_1 \) and \( \phi_2 \) be the angle of inclination of the second line with slope \( m_2 \). Therefore, we have:
\[ m_1 = \tan(\phi_1) \quad \text{and} \quad m_2 = \tan(\phi_2) \]

3. **Finding the Angle Between the Lines**:
The angle \( \theta \) between the two lines can be expressed as:
\[ \theta = \phi_2 - \phi_1 \]

4. **Using the Tangent of the Angle Difference**:
We can use the formula for the tangent of the difference of two angles:
\[ \tan(\theta) = \tan(\phi_2 - \phi_1) = \frac{\tan(\phi_2) - \tan(\phi_1)}{1 + \tan(\phi_1) \tan(\phi_2)} \]
Substituting the values of \( m_1 \) and \( m_2 \):
\[ \tan(\theta) = \frac{m_2 - m_1}{1 + m_1 m_2} \]

5. **Considering the Absolute Value**:
Since we are interested in the acute angle, we take the absolute value:
\[ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]

6. **Conclusion**:
The given statement claims that the acute angle between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]
This matches our derived formula, therefore the statement is **True**.

### Final Answer:
**True**