We want to determine which of the given statements is true when parallel lines are cut by a transversal. We will analyze these statements one at a time.

Statement A

Let's consider statement A.

Vertical angles are supplementary.

Recall that when two lines or line segments intersect, vertical angles are formed on opposite sides of the point of intersection. Vertical angles have always equal measures, but they do not have to be supplementary. Let's look at an example of two lines cut by a transversal.

In this diagram, we can see two vertical angles and the sum of their measures is not equal to

18 0^{\circ} .

14 0^{\circ} + 14 0^{\circ} = 28 0^{\circ}

Therefore, these angles are not supplementary. This means that statement A is not true.

Statement B

Now we will analyze statement B.

Alternate exterior angles are supplementary.

Alternate exterior angles are angles exterior to the two lines cut by the transversal that lie on opposite sides of it. When the lines are parallel, their measures are equal, but they do not have to be supplementary. Let's return to our example of two lines cut by a transversal.

In this diagram, we can see two labeled alternate exterior angles. The sum of their measures is not equal to

18 0^{\circ} .

14 0^{\circ} + 14 0^{\circ} = 28 0^{\circ}

Therefore, these angles are not supplementary, so statement B is not true.

Statement C

Next, we will consider statement C.

Alternate interior angles are supplementary.

Recall that alternate interior angles are interior angles that lie on opposite sides of the transversal. When the lines are parallel, their measures are equal. However, alternate interior angles do not have to be supplementary. Consider our example diagram again.

Two alternate interior angles are labeled this time. The sum of their measures is not equal to

18 0^{\circ} .

14 0^{\circ} + 14 0^{\circ} = 28 0^{\circ}

Therefore, these angles are not supplementary. This means that statement C is not true.

Statement D

Finally, we will analyze statement D.

Corresponding angles have the same measure.

Corresponding angles are angles that are in the same position on the two lines in relation to the transversal. When the lines are parallel, their measures are equal. Therefore, statement D is true when parallel lines are cut by a transversal.

Conclusion

We found that only statement D is true.

Exercise