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Proving Lines are Perpendicular, a la Shmoop.

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Leroy, the city pigeon, is having a bit of a tiff with his country cousin, Roy Lee.

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Leroy argues that the streets below form a perfect cross…

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…but Roy Lee insists that the streets look kind of… slanted.

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How can Leroy prove that the streets do, in fact, form a perfect cross,

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and that they’re perpendicular?

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First let’s think about the streets as two intersecting lines on a map.

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Then we can use some useful theorems that will help us determine if the lines are, in

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fact, perfectly perpendicular.

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If two lines are perpendicular, they’ll form four right angles at their intersection.

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Right angles measure 90 degrees, and it appears that Leroy’s two streets create four right

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angles at their intersection.

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But Leroy wants to make sure.

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Another theorem Leroy can use follows the same principle:

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It says “If two lines intersect to form a linear pair of congruent angles,

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then the lines are perpendicular.”

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Angles 1 and 4 are adjacent and form a linear pair.

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Two angles in a linear pair are always supplementary,

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which means that their measures must add to 180 degrees.

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That means the measure of angle 1 plus the measure of angle 4 equals 180 degrees.

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If these angles have to be congruent, their measures should to be equal to each other.

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So let's substitute the measure of angle 4 with the measure of angle 1.

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They're both equal anyway, right?

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We can add the measure of angle 1 and the measure of angle 1 to get 2 times the measure of angle 1.

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To isolate the measure of angle 1, we can divide 2 by both sides

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and get that the measure of angle 1 is 90 degrees.

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Because the two angles are congruent, we know that 90 degrees must also be the measure of angle 4.

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Since our 90 degree angles are a linear pair of congruent angles, like the theorem says,...

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...we know the lines are perpendicular. Voila!

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He even uses a protractor to be certain.

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Looks like this theorem reinforces Leroy’s argument.

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Leroy was indeed correct… the streets ARE perpendicular.

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Oh, great. Now he’s going to do that obnoxious “victory strut” stuff.

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