Which angles are equal to each other?

The image shows intersecting lines, forming angles labeled 1, 2, 3, and 4. We are given the equation \(\angle 1 - \angle 2 = 120^{\circ}\) and asked to find \(\angle 3\) and \(\angle 4\).

Angles 1 and 3 are vertically opposite angles. Vertically opposite angles are equal.
Therefore, \(\angle 1 = \angle 3\).

Angles 2 and 4 are also vertically opposite angles.
Therefore, \(\angle 2 = \angle 4\).

Also, angles 1 and 2 form a linear pair, meaning they are adjacent angles on a straight line. The sum of angles in a linear pair is 180 degrees.
So, \(\angle 1 + \angle 2 = 180^{\circ}\).

We are given \(\angle 1 - \angle 2 = 120^{\circ}\).
We have a system of two equations:
1. \(\angle 1 + \angle 2 = 180^{\circ}\)
2. \(\angle 1 - \angle 2 = 120^{\circ}\)

Adding equation 1 and equation 2:
\((\angle 1 + \angle 2) + (\angle 1 - \angle 2) = 180^{\circ} + 120^{\circ}\)
\(2\angle 1 = 300^{\circ}\)
\(\angle 1 = \frac{300^{\circ}}{2}\)
\(\angle 1 = 150^{\circ}\)

Now, substitute the value of \(\angle 1\) into equation 1:
\(150^{\circ} + \angle 2 = 180^{\circ}\)
\(\angle 2 = 180^{\circ} - 150^{\circ}\)
\(\angle 2 = 30^{\circ}\)

Since \(\angle 1 = \angle 3\), then \(\angle 3 = 150^{\circ}\).
Since \(\angle 2 = \angle 4\), then \(\angle 4 = 30^{\circ}\).

Answer: \(\angle 1 = \angle 3 = 150^{\circ}\) and \(\angle 2 = \angle 4 = 30^{\circ}\).