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Parallel and Perpendicular Lines

Parallel and Perpendicular Lines

At a Glance - Perpendicular Lines

From classics like Die Hard and Terminator to the legendary Kung Pow: Enter the Fist, action movies have provided countless hours of bated breath, wide-eyed stares, and unrealistic maneuvers that later become the subject of MythBusters episodes. (Ever wondered if it's actually possible to outfox a laser beam alarm system? Find out!) What beats a bucket of buttery popcorn and all the hand-to-hand combat, ginormous explosions, and obligatory car chase scenes that only action movies can provide?

Well, geometry doesn't have car chase scenes per se, but head-on crashes are far more common than you might think. You never expected math to be this action-packed, did you?

In geometry, we call two lines that crash into each other perpendicular, which is kind of the opposite of parallel. Not only do perpendicular lines intersect, they really, really intersect. We're talking a full-on crash that makes not one, not two, not even three, but four right angles.

Perpendicular lines make four right angles? Proving it is elementary, my dear Watson.

Sample Problem

Given: Lines m and n are perpendicular.
Prove: ∠1, ∠2, ∠3, and ∠4 are right angles.

1. Lines m and n are perpendicularGiven
2. ∠1 is a right angleDefinition of perpendicular
3. m∠1 = 90Definition of right angle
4. ∠1 ≅ ∠3Vertical angles theorem
5. m∠3 = 90Definition of congruent angles, substitution
6. ∠1 and ∠4 are supplementaryGiven in figure, straight angle theorem
7. m∠1 + m∠4 = 180Definition of supplementary angles
8. m∠4 = 90Substitution of (3) into (7)
9. ∠4 ≅ ∠2Vertical angles theorem
10. m∠2 = 90Definition of congruent angles, substitution
11. ∠1, ∠2, ∠3, and ∠4 are right anglesDefinition of right angle using (3), (5), (8), and (10)

So when two perpendicular lines intersect, we got something that looks like this.

Marking every angle with a 90° or right angle box is a little redundant though, so we usually just mark one of them. That's enough to know the rest of them are right angles, too.

Parallel lines get their own symbol, so it only makes sense that perpendicular lines get their own symbol too. We denote that lines FR and ED are perpendicular by writing FRED.

Perpendicular segments also have the peculiar quality of being the shortest distances from any point to another line. So if you're standing 20 feet away from a wall, that measurement is along a segment that's perpendicular to the wall.

It's also important to know that if two lines are both perpendicular to a third line, then those two lines are parallel to each other. So if lines f and g both intersect line b at 90° angles, then f and g are parallel. You can use your knowledge of corresponding angles, alternate interior angles, or alternate exterior angles to prove this.

Also, make sure you know that orthogonal means the same thing as perpendicular. Some teachers or textbooks like to use it because it makes them sound really fancy, like using the word "loquacious" instead of "talkative." They can use all the five-dollar words they want. We'll stick to ones that only cost a buck fifty and get a side of fries with the extra change.

Example 1

If two lines intersect and one of the angles formed from the intersection is 90°, what is the measurement of the adjacent angle?

Example 2

Which of the following lines are perpendicular?

Example 3

What is the relationship of BG and CF?

Example 4

Find a line that's perpendicular to 3x + 2y = 10 and passes through the origin.

Exercise 1

Is line p perpendicular to line r?

Exercise 2

Is line q perpendicular to line r?

Exercise 3

Which of these lines are parallel and which are perpendicular?


Exercise 4

What is the measure of ∠RZQ?

Exercise 5

If the measure of ∠UYZ is 19°, what is the measure of ∠WYP?


Exercise 6

Which lines are parallel? Which lines are perpendicular?

Exercise 7

The equation of line a is y = -⅕x – 1. If line b is perpendicular to line a and intersects line a when x = 5, what is the equation of b?

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