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 use the formula for the angle between two lines. If the equations of the lines are \(y = m_1x + c_1\) and \(y = m_2x + c_2\), then the angle \(\theta\) between the two lines can be calculated using the following formula:
\(\theta = \tan^{-1}\left(\frac{|m_2 - m_1|}{1 + m_1 m_2}\right)\)
Here, \(m_1 = 2 - \sqrt{3}\) and \(m_2 = 2 + \sqrt{3}\). Let's plug these values into the formula:
\(\theta = \tan^{-1}\left(\frac{|(2 + \sqrt{3}) - (2 - \sqrt{3})|}{1 + (2 - \sqrt{3})(2 + \sqrt{3})}\right)\)
Simplifying the numerator:
\(= \tan^{-1}\left(\frac{|2 + \sqrt{3} - 2 + \sqrt{3}|}{1 + (2^2 - (2 \times 2\sqrt{3}) + (\sqrt{3})^2)}\right)\)
\(= \tan^{-1}\left(\frac{2\sqrt{3}}{1 + (4 - 4 + 3)}\right)\)
\(= \tan^{-1}\left(\frac{2\sqrt{3}}{1 + 3}\right)\)
\(= \tan^{-1}\left(\frac{2\sqrt{3}}{4}\right)\)
\(= \tan^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
We know from trigonometry that:
\(\tan(60^\circ) = \sqrt{3}\), therefore the angle between the two lines is:
\(\theta = 60^\circ\)
Thus, the correct answer is \(60^\circ\).