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Proof: parallel lines have the same slope

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

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  • leaf green style avatar for user Kianna Marquez
    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?
    (8 votes)
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  • blobby green style avatar for user Marie Wadsworth
    Is this material on the GED test? I teach GED classes to adults. I also was curious of its applications in the world. Thanks!
    (5 votes)
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  • eggleston green style avatar for user Ants Work Diligently
    Couldn't the slope be negative, and still have the same ratios of sides of right triangles drawn on them?
    (3 votes)
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  • purple pi teal style avatar for user Thunder_Coder ⚡
    Does this have to do with trigonometry?
    cause if so it explains why i didn't understand this
    (4 votes)
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  • blobby green style avatar for user Aditya Nagesh
    Sal could use a vector version to generalize the relations between parallel and perpendicular lines.
    (2 votes)
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  • hopper jumping style avatar for user kyle shapiro
    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!
    (2 votes)
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    • duskpin sapling style avatar for user proxima
      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.
      (2 votes)
  • male robot hal style avatar for user eakpainyang2
    Why didn't he say the AEC is a 90 degree angle?
    (2 votes)
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  • leafers seedling style avatar for user PAINT
    I think for math problems like theses nothing can be proven. In some problems I found out parallel lines fo NOT have the same slope using triangle similarity. so....?
    (0 votes)
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  • leafers sapling style avatar for user tømcat
    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 :)
    (1 vote)
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    • leaf green style avatar for user kubleeka
      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)
  • leaf blue style avatar for user Smit
    When you set up the ratios, do you always put the shorter side on the bottom?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      Sal's ratios represent the slope of each line. Slope is always "change in Y / Change in X". So, the top number needs to be the length of the vertical side and the bottom number needs to be the length of the horizontal side.

      I'm not sure why you said the shorter side is on the bottom. The shorter sides are actually the vertical sides and they are on top.
      (1 vote)

Video transcript

- [Voiceover] What I wanna do in this video is prove that parallel lines have the same slope. So let's draw some parallel lines here. So, that's one line and then let me draw another line that is parallel to that. I'm claiming that these are parallel lines. And now, I'm gonna draw some transversals here. So first let me draw a horizontal transversal. So, just like that. And then let me do a vertical transversal. So, just like that. And I'm assuming that the green one is horizontal and the blue one is vertical. So we assume that they are perpendicular to each other, that these intersect at right angles. And from this, I'm gonna figure out, I'm gonna use some parallel line angle properties to establish that this triangle and this triangle are similar and then use that to establish that both of these lines, both of these yellow lines have the same slope. So actually let me label some points here. So let's call that point A, point B, point C, point D, and point E. So, let's see. First of all we know that angle CED is going to be congruent to angle AEB, because they're both right angles. So that's a right angle and then that is a right angle right over there. We also know some things about corresponding angles for where our transversal intersects parallel lines. This angle corresponds to this angle if we look at the blue transversal as it intersects those two lines. And so they're going to be, they're going to have the same measure, they're going to be congruent. Now this angle on one side of this point B is going to also be congruent to that, because they are vertical angles. We've seen that multiple times before. And so we know that this angle, angle ABE is congruent to angle ECD. Sometimes this is called alternate interior angles of a transversal and parallel lines. Well, if you look at triangle CED and triangle ABE, we see they already have two angles in common, so if they have two angles in common, well, then their third angle has to be in common. So, because this third angle's just gonna be 180 minus these other two, and so this third angle is just gonna be 180 minus this, the other two. And so just like that, we notice we have all three angles are the same in both of these triangles, well, they're not all the same, but all of the corresponding angles, I should say, are the same. This blue angle has the same measure as this blue angle, this magenta angle has the same measure as this magenta angle, and then the other angles are right angles, these are right triangles here. So we could say triangle AEB, triangle AEB is similar, similar similar to triangle DEC, triangle DEC by, and we could say by angle, angle, angle, all the corresponding angles are congruent, so we are dealing with similar triangles. And so we know similar triangles are a ratio of corresponding sides are going to be the same. So we could say that the ratio of let's say the ratio of BE, the ratio of BE, let me write this down, this is this side right over here, the ratio of BE to AE, to AE, to AE, is going to be equal to, so that side over that side, well what is the corresponding side? The corresponding side to BE is side CE. So that's going to be the same as the ratio between CE and DE, and DE. And this just comes out of similar, the similarity of the triangles, CE to DE. So once again, once we established these triangles are similar, we can say the ratio of corresponding sides are going to be the same. Now what is the ratio between BE and AE? The ratio between BE and AE. Well that is the slope of this top line right over here. We could say that's the slope of line AB, slope of line connecting, connecting A to B. All right, let me just use, I could write it like this, that is slope of, slope of A, slope of line AB. Remember slope is, when you're going from A to B, it's change in y over change in x. So when you're going from A to B, your change in x is AE, and your change in y is BE, or EB, however you want to refer to it. So this right over here is change in y, and this over here is change in x. Well, now let's look at this second expression right over here, CE over DE, CE over DE. Well, now, this is going to be change in y over change in x between point C and D. So this is, this right over here, this is the slope of line, of line CD. And so just like that, by establishing similarity, we were able see the ratio of corresponding sides are congruent, which shows us that the slopes of these two lines are going to be the same. And we are done.