How can one prove that two lines are parallel?

There are four major ways of proving if two lines are parallel.

1. Corresponding Angles Postulate

If you read the previous post then you already know that corresponding angles are angles which are formed when a transversal crosses two other lines. 

The following are corresponding angles:

1 and 5

2 and 6

3 and 7

4 and 8

Knowing corresponding angles is very useful to make sure two angles are parallel, for if the corresponding angles have the same measurement (degree), then the two lines being crossed by the transversal are parallel. If their measurement are different then, no, they are not parallel.   

2.  Alternate Interior Angles Theorem

3. Alternate Exterior Angles Theorem  
    

The following are alternate exterior angles: 

8 and 1

7 and 2

The following are alternate interior angles:

6 and 3

5 and 4 

The previous post also talked about alternate interior and exterior angles theorem. This is useful to figure out if two lines are parallel because if the alternate interior angles are congruent (have the same measurement), then the two lines are automatically parallel. Same thing goes with the alternate exterior angles; if they are congruent, then the two lines being intersected by the transversal are parallel. 

4. Consecutive Interior Angles Theorem 

Consecutive angles, are angles which lay on the same side of the transversal and are on the inside of the two lines.

The following are consecutive interior angles: 

6 and 4

3 and 5

This theorem is useful to find out if two lines are parallel because if both the consecutive interior angles' measurement add up to 180, then the two lines are parallel.