We are asked to explain how we can use angle measures to tell whether two lines are parallel. To do so, we use the relationship between the angles created when a transversal cuts the two lines. Let's consider two example lines $a$ and $b .$

Let's cut the two lines with a transversal. This creates $8$ angles.

When two lines are cut by a transversal, we can use three different criteria to tell whether the lines are parallel.

If the corresponding angles are congruent, the lines are parallel.
If the alternate interior angles are congruent, the lines are parallel.
If the same-side interior angles are supplementary, the lines are parallel.

Let's use each of these rules to confirm whether our two lines are parallel. First, we will check whether the same-side interior angles are supplementary. In our case, the pairs of same-side interior angles are $∠ 4$ and $∠ 5,$ and $∠ 3$ and $∠ 6 .$

Let's note the measures of these angles.

As we can see, in both cases the sum of the measures of the angles is

18 0^{\circ} .

8 3^{\circ} + 9 7^{\circ} = 18 0^{\circ}

This means that same-side interior angles are supplementary, so the lines are parallel. Now let's confirm that the lines are parallel using alternate interior angles. The pairs of alternate interior angles are

∠ 4

and

∠ 6,

and

∠ 3

and

∠ 5 .

Once again, we note the measures of these angles.

As we can see, both pairs of alternate interior angles are congruent. This confirms that the lines are parallel. Finally, we will use the angle measures of corresponding angles. Here, the pairs of corresponding angles are $∠ 1$ and $∠ 5,$ $∠ 2$ and $∠ 6,$ $∠ 3$ and $∠ 7,$ and $∠ 4$ and $∠ 8 .$

Note the measures of these angles.

All corresponding angles are congruent, which means that the lines are parallel.

Exercise