Parallel Lines: How to Prove Lines Are Parallel

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  • 0:06 Proving Parallel Lines
  • 1:44 Corresponding Angles
  • 2:36 Alternate Interior Angles
  • 3:44 Alternate Exterior Angles
  • 4:39 Supplementary Angles
  • 5:40 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can prove that two lines are parallel just by matching up pairs of angles. Learn which angles to pair up and what to look for.

Proving Parallel Lines

Picture a railroad track and a road crossing the tracks. You know that the railroad tracks are parallel; otherwise, the train wouldn't be able to run on them without tipping over. But, how can you prove that they are parallel? First, you recall the definition of parallel lines, meaning they are a pair of lines that never intersect and are always the same distance apart. Then you think about the importance of the transversal, the line that cuts across two other lines. At this point, you link the railroad tracks to the parallel lines and the road with the transversal. Now what?

Now you get to look at the angles that are formed by the transversal with the parallel lines. There are four different things you can look for that we will see in action here in just a bit. Just remember that when it comes to proving two lines are parallel, all you have to look at are the angles. Specifically, we want to look for pairs of:

  1. Corresponding angles
  2. Alternate interior angles
  3. Alternate exterior angles, or
  4. Supplementary angles

If we find just one pair that works, then we know that the lines are parallel. Also, you will see that each pair has one angle at one intersection and another angle at another intersection. When I say intersection, I mean the point where the transversal cuts across one of the parallel lines. So, since there are two lines in a pair of parallel lines, there are two intersections.

Corresponding Angles

Corresponding angles are the angles that are at the same corner at each intersection. This means that if my first angle is at the top left corner of one intersection, the matching angle at the other intersection is also at the top left. Since there are four corners, we have four possibilities here:

If the corresponding angles have the same measurement, the lines are parallel.
image showing lines and corresponding angles

We can match the corners at top left, top right, lower left, or lower right.

What we are looking for here is whether or not these two angles are congruent or equal to each other. If they are, then the lines are parallel. So, if both of these angles measured 60 degrees, then you know that the lines are parallel. But, if the angles measure differently, then automatically, these two lines are not parallel.

Alternate Interior Angles

Alternate interior angles is the next option we have. These angles are the angles that are on opposite sides of the transversal and inside the pair of parallel lines. So, you will have one angle on one side of the transversal and another angle on the other side of the transversal. And, both of these angles will be inside the pair of parallel lines. The inside part of the parallel lines is the part between the two lines. So, for the railroad tracks, the inside part of the tracks is the part that the train covers when it goes over the tracks. We have two possibilities here:

If the alternate interior angles are equal, the lines are parallel.
image showing lines and alternate interior angles

We can match top inside left with bottom inside right or top inside right with bottom inside left.

Also here, if either of these pairs is equal, then the lines are parallel. So, say the top inside left angle measures 45, and the bottom inside right also measures 45, then you can say that the lines are parallel.

Alternate Exterior Angles

Next is alternate exterior angles. These are the angles that are on opposite sides of the transversal and outside the pair of parallel lines. Going back to the railroad tracks, these pairs of angles will have one angle on one side of the road and the other angle on the other side of the road. But, both of these angles will be outside the tracks, meaning they will be on the part that the train doesn't cover when it goes over the tracks. We also have two possibilities here:

If either of the alternate exterior angles are equal, the lines will be parallel.
image showing lines with alternate exterior angles

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