Use a paragraph proof to prove:

if two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel.

To prove that if two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel, you can use a paragraph proof. Here's how you can approach it:

Let's assume we have two lines, line a and line b, and a transversal, line t, as shown in the diagram.

1. Given: Line a is intersected by line t, and line b is also intersected by line t.
2. Assume that the corresponding angles formed by line a and line t, and line b and line t are congruent.
- Let's call the corresponding angles angle 1 and angle 2.
- So, angle 1 ≅ angle 2.

Now, we will proceed with the proof by contradiction:

3. Assume that lines a and b are not parallel.
4. If lines a and b are not parallel, then they intersect at some point, say point P.
5. As lines a and b intersect at point P, they form vertical angles.
6. By the Vertical Angles Theorem, vertical angles are congruent.
- Therefore, the vertical angles formed by lines a and b at point P are congruent.

But, since we assumed that the corresponding angles formed by line a and line t, and line b and line t are congruent (angle 1 ≅ angle 2), and vertical angles are congruent, it means that angle 1 ≅ angle 2 ≅ vertical angles formed by lines a and b.

This contradicts our assumption that lines a and b are not parallel. By contradiction, we conclude that if the corresponding angles formed by two lines and a transversal are congruent, then the two lines must be parallel.