Perpendicular Lines Examples

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

It's your lucky day, because you've got two options here. Yes, that's right: there are two ways to figure this problem out.

First, we know that if two lines intersect, one angle and its adjacent angle will add up to 180° because they form a straight angle together. If one is 90°, the other is 180° – 90° = 90°. No, that's not a crazy coincidence. Why? Because of our second option.

When two lines form a 90° angle, they're perpendicular. Perpendicular lines actually form four 90° angles, not just one. So the adjacent angle to a 90° angle will be a 90° angle also. Shocker.

Which of the following lines are perpendicular?

When two perpendicular lines love each other very much and are ready to commit to one another, they have intersects and nine months later, a 90° angle is formed. Actually, they have quadruplets. Ouch.

What we're saying is that the right angle is the offspring of perpendicular lines. That means BG is perpendicular to HE because there is a right angle between them. Of course, CF ⊥ HE for the same reason.

What is the relationship of BG and CF?

Lines BG and CF aren't perpendicular to each other. They're both perpendicular to a third line. What type of relationship does that give them?

You may see it intuitively, but how can we explain this mysterious phenomenon? Look at the corresponding angles (∠HLI and ∠LKJ). They're congruent. So are alternate interior angles (∠LKJ and ∠KLG) and alternate exterior angles (∠HLI and ∠FKE). By all accounts, lines BG and CF are parallel.

We can explain this using slope as well. If HE has a slope of m, we know that both BG and CF need to have a slope that's the opposite reciprocal of m in order to be perpendicular. Since they both have the same slope (-¹⁄_m), they're both parallel to each other.

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

Get your skis and grab your goggles—we're about to hit the slopes!

Rearranging 3x + 2y = 10 into slope-intercept form gives us y = -³/₂x + 5. The negative reciprocal of -³/₂ is ⅔, so we're looking for an equation that takes the form y = ⅔x + b. All that's left to do is find b.

Since we know this line passes through the origin, we can plug in y = 0 and x = 0 to find b.

y = ⅔x + b
0 = ⅔(0) + b
0 = 0 + b
b = 0

Hopefully, that's not a huge surprise. The equation of our line is y = ⅔x.