Find the angle between the lines `y=(2-sqrt3)(x+5)` and `y=(2+sqrt3)(x-7)`.

Explanation

To find the angle between the two lines given by the equations \( y = (2 - \sqrt{3})(x + 5) \) and \( y = (2 + \sqrt{3})(x - 7) \), we can follow these steps:

### Step 1: Identify the slopes of the lines
The equations of the lines can be rewritten in slope-intercept form \( y = mx + c \), where \( m \) is the slope.

1. For the first line:
\[ y = (2 - \sqrt{3})(x + 5) \]
Expanding this:
\[ y = (2 - \sqrt{3})x + (2 - \sqrt{3}) \cdot 5 \]
The slope \( m_1 \) is:
\[ m_1 = 2 - \sqrt{3} \]

2. For the second line:
\[ y = (2 + \sqrt{3})(x - 7) \]
Expanding this:
\[ y = (2 + \sqrt{3})x - (2 + \sqrt{3}) \cdot 7 \]
The slope \( m_2 \) is:
\[ m_2 = 2 + \sqrt{3} \]

### Step 2: Use the formula for the angle between two lines
The formula for the tangent of the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

### Step 3: Calculate \( m_1 - m_2 \) and \( m_1 m_2 \)
1. Calculate \( m_1 - m_2 \):
\[ m_1 - m_2 = (2 - \sqrt{3}) - (2 + \sqrt{3}) = -2\sqrt{3} \]

2. Calculate \( m_1 m_2 \):
\[ m_1 m_2 = (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 4: Substitute into the formula
Substituting the values into the formula:
\[ \tan \theta = \left| \frac{-2\sqrt{3}}{1 + 1} \right| = \left| \frac{-2\sqrt{3}}{2} \right| = \left| -\sqrt{3} \right| = \sqrt{3} \]

### Step 5: Find the angle \( \theta \)
Now, we know that:
\[ \tan \theta = \sqrt{3} \]
The angles that satisfy this equation are:
\[ \theta = 60^\circ \quad \text{or} \quad \theta = 120^\circ \]

### Final Answer
The angle between the lines is either \( 60^\circ \) or \( 120^\circ \).

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