Proving Lines are Parallel

To prove lines are parallel, you need a third line. We at Shmoop (and the rest of the world) call it a transversal.

ConstructionParallel Lines
GeometryParallel and Perpendicular Lines
LanguageEnglish Language
Mathematics and Statistics AssessmentPlane Geometry
Parallel and Perpendicular LinesParallel Lines
Plane GeometryPlane figures (triangles, rectangles, parallelograms, trapezoids, circles, polygons)

Transcript

00:26

But hang on a sec; do they really have to be worried in the first place?

00:32

By definition, parallel lines never cross, even if they go on forever.

00:36

If we can prove that these snails’ paths are parallel to each other…

00:39

…then they can stop arguing and leave the forest undergrowth in peace.

00:43

We’ve got several important theorems and postulates to help us out.

00:47

All of those theorems and postulates have to do with transversals…

00:51

…a third line cutting across the two lines in question.

00:56

A straight branch lies across the snails’ paths… that can serve as our transversal.

01:02

A transversal will create 8 different angles we should examine before reaching our conclusion.

01:09

The first postulate we can use to determine whether two lines are parallel is the Corresponding

01:18

Angles Converse, which states that…

01:25

…“If two lines are cut by a transversal so that corresponding angles are congruent,

01:30

then the lines are parallel.”

01:39

Remember that corresponding angles refer to angles such as 2 and 6, or 3 and 7.

01:45

Check. The first theorem we can use is the Alternate

01:45

Exterior Angles Converse, which states…

01:46

…“If two lines are cut by a transversal so that alternate exterior angles are congruent,

01:52

then the lines are parallel.”

01:53

Remember that alternate exterior angles refer to angles such as 2 and 8, or 1 and 7.

01:56

Check. The second theorem we can use is the Consecutive

01:59

Interior Angles Converse, which states…

02:02

…“If two lines are cut by a transversal so that consecutive interior angles are supplementary…

02:11

meaning they add up to 180 degrees… then the lines are parallel.”

02:17

Consecutive interior angles, in this case, refer to angles 3 and 6.

02:20

The branch creates consecutive interior angles measuring 150 degrees and 30 degrees; 150

02:28

plus 30 is 180.

02:32

Check. The last theorem we can use is the Alternate

02:40

Interior Angles Converse, which states…

02:46

…“If two lines are cut by a transversal so that alternate interior angles are congruent,

02:52

then the lines are parallel.”

02:54

Alternate Interior Angles refer to angles 4 and 6 or 3 and 5.

02:54

Check-a-rooni. It seems those snails don’t have to worry

02:58

about crashing into each other after all.

03:00

But they just might run into a few more transversals on their journey.