If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## High school geometry

### Course: High school geometry>Unit 4

Lesson 6: Proving relationships using similarity

# Proof: parallel lines have the same slope

Sal proves that parallel lines have the same slope using triangle similarity.

## Want to join the conversation?

• At , how can you assume that the lines which the vertical transversal passes through are parallel? I know that you claim that they are in the beginning, but what if that information is not given in a problem?
• If the lines are parallel, certainly the exercise will tell you.
• Does this have to do with trigonometry?
cause if so it explains why i didn't understand this
• No this is not trigonometry. You can think of this as Algebra I or pre-algebra. They just wanted to show why this is always true
• Is this material on the GED test? I teach GED classes to adults. I also was curious of its applications in the world. Thanks!
• in the world there are many times when proving to things are parallel is vital(this being basiclly the idea in the video, even if not said directly):
-archetecture
-mathmatician
engineer
botanist
plastic surgeon
8 year old you likes to play minecraft pocket edition on his moms phone whil wait in the grocery store line.
and many more😉
• Sal making me think I had a dead pixel for a moment there.
• Couldn't the slope be negative, and still have the same ratios of sides of right triangles drawn on them?
• Yes, if the right end of the parallel lines had been drawn angled down their slopes would have been negative.
• Sal could use a vector version to generalize the relations between parallel and perpendicular lines.
• what i don't get about this is that he's proving the fact that they have the same slope by the fact that he has the same slope.... isn't that the definition of a parallel line ?

sorry if i'm misunderstanding this. help is appreciated :)
• Two lines are parallel if they lie in the same plane and don't intersect. The definition has nothing to do with coordinate geometry or slopes.

By proving that parallel lines have the same slope, Sal is translating the concept of parallelism from synthetic (non-coordinate) geometry to coordinate geometry.
(1 vote)
• I feel like I'm not understanding the value of this lesson. It feels like there's no value to this proof. Let's substitute language to try and illustrate my point. "Parallel lines have the same slope". What does "parallel" mean? To me, "parallel" might be defined as "two lines that don't overlap and have the same slope". So if we substitute that back in, we get "two lines that don't overlap and have the same slope lines have the same slope". It's like proving that a chicken is a bird. Chicken is the subject, bird is the category. By definition a chicken is a bird; Why would we try proving an already agreed upon definition? Is the point of this exercise to draw a connection between linear algebra and geometry? Is it a "true math proof" if the arguments contain the conclusion? Or is my definition for the word "parallel" different from other people? Any assistance would be appreciated. Thank you!
• Your definition of parallel is exactly right, but think back to a long time ago when people didn't know the exact definition of parallel. It was not something that was already defined so people had to prove that parallel lines have the same slope. The reason we know the that parallel lines have the same slope is because of this proof. Sometimes, on a test, you might also be asked to prove the same thing.