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Proving Lines are Parallel, a la Shmoop.

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On the forest floor, tensions are brewing.

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It seems two snails are in the midst of a heated argument.

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They are worried that their paths on the forest floor may cross; if that happens, they might

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crash. And they just let their insurance lapse.

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

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

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

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

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

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

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

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

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

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

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Angles Converse, which states that…

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

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then the lines are parallel.”

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

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Check. The first theorem we can use is the Alternate

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Exterior Angles Converse, which states…

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

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then the lines are parallel.”

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

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Check. The second theorem we can use is the Consecutive

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Interior Angles Converse, which states…

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

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

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

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

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plus 30 is 180.

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Check. The last theorem we can use is the Alternate

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Interior Angles Converse, which states…

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

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then the lines are parallel.”

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

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

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about crashing into each other after all.

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

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