find the angle between the pair of lines `(x+3)/3=(y-1)/5=(z+3)/4` and `(x+1)/1=(y-4)/1=(z-5)/2`

Explanation

To find the angle between the given pair of lines, we can follow these steps:

### Step 1: Identify the direction ratios of the lines
The equations of the lines are given in symmetric form. We can extract the direction ratios from these equations.

For the first line:
\[ \frac{x + 3}{3} = \frac{y - 1}{5} = \frac{z + 3}{4} \]
The direction ratios (denoted as \( l_1, m_1, n_1 \)) for the first line are \( (3, 5, 4) \).

For the second line:
\[ \frac{x + 1}{1} = \frac{y - 4}{1} = \frac{z - 5}{2} \]
The direction ratios (denoted as \( l_2, m_2, n_2 \)) for the second line are \( (1, 1, 2) \).

### Step 2: Use the formula for the angle between two lines
The cosine of the angle \( \theta \) between two lines with direction ratios \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \) is given by the formula:
\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}} \]

### Step 3: Calculate the numerator
Substituting the direction ratios into the formula:
\[ \cos \theta = \frac{(3)(1) + (5)(1) + (4)(2)}{\sqrt{3^2 + 5^2 + 4^2} \sqrt{1^2 + 1^2 + 2^2}} \]
Calculating the numerator:
\[ = 3 + 5 + 8 = 16 \]

### Step 4: Calculate the denominator
Now we calculate the denominator:
\[ \sqrt{3^2 + 5^2 + 4^2} = \sqrt{9 + 25 + 16} = \sqrt{50} \]
\[ \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \]
Thus, the denominator becomes:
\[ \sqrt{50} \cdot \sqrt{6} = \sqrt{300} = 10\sqrt{3} \]

### Step 5: Substitute back into the cosine formula
Now substituting back into the cosine formula:
\[ \cos \theta = \frac{16}{10\sqrt{3}} = \frac{8}{5\sqrt{3}} \]

### Step 6: Find the angle \( \theta \)
To find the angle \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{8}{5\sqrt{3}}\right) \]

### Final Answer
Thus, the angle between the given pair of lines is:
\[ \theta = \cos^{-1}\left(\frac{8}{5\sqrt{3}}\right) \]