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

Construction | Parallel Lines |

Geometry | Parallel and Perpendicular Lines |

Language | English Language |

Mathematics and Statistics Assessment | Plane Geometry |

Parallel and Perpendicular Lines | Parallel Lines |

Plane Geometry | Plane figures (triangles, rectangles, parallelograms, trapezoids, circles, polygons) |

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

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

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

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

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

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

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

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

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

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

Angles Converse, which states that…

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

then the lines are parallel.”

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

Check. The first theorem we can use is the Alternate

Exterior Angles Converse, which states…

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

then the lines are parallel.”

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

Check. The second theorem we can use is the Consecutive

Interior Angles Converse, which states…

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

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

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

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

plus 30 is 180.

Check. The last theorem we can use is the Alternate

Interior Angles Converse, which states…

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

then the lines are parallel.”

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

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

about crashing into each other after all.

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