From the image, solve the problem for \(\angle\) 1 and \(\angle\) 2 where \(\angle\) 1 : \(\angle\) 2 = 3 : 1.

Solution:

We are given the ratio \( \angle 1 : \angle 2 = 3 : 1 \).

From the diagram, lines 'm' and 'n' are parallel, and they are intersected by a transversal. \(\angle 1\) and \(\angle 2\) are consecutive interior angles.

Consecutive interior angles are supplementary, meaning their sum is \( 180^{\circ} \). Therefore, \( \angle 1 + \angle 2 = 180^{\circ} \).

Let \( \angle 1 = 3x \) and \( \angle 2 = 1x \) (or simply \( x \)) based on the given ratio.

Substituting these into the equation for supplementary angles:

\( 3x + x = 180^{\circ} \)

\( 4x = 180^{\circ} \)

Divide both sides by 4:

\( x = \frac{180^{\circ}}{4} \) \( x = 45^{\circ} \)

Now, we can find the values of \(\angle 1\) and \(\angle 2\):

\( \angle 1 = 3x = 3 \times 45^{\circ} = 135^{\circ} \)

\( \angle 2 = x = 45^{\circ} \)

Let's verify: \( \angle 1 + \angle 2 = 135^{\circ} + 45^{\circ} = 180^{\circ} \). The ratio is \( 135 : 45 = 3 : 1 \).

Answer: \(\angle 1 = 135^{\circ}\), \(\(\angle\) 2 = 45^{\(\circ\)}\}.