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# Proof: parallel lines have the same slope

CCSS.Math:

## Video transcript

what I want to 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 going to 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 going to figure out I'm going to use I'm going to use some parallel line angle properties to establish that this triangle and this triangle are similar and then use that to establish that the both of these lines both of these yellow lines have the same slope so Lexi 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 a B a B 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 transverse where 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 convert to that because they are vertical angles and we've seen that multiple times before and so we know that this angle angle a B E is congruent to angle E CD sometimes this is called alternate interior angles of a transversal and parallel lines well if you look at triangle CED and triangle a B II we see they already have two angles in common so if they have two angles in common well their third angle has to be in common so because this third angles just going to be 180 minus these other two and so this third angle just going to 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 whether they're not all the same but all of the corresponding angles I should say are the same angle angle this blue angle is the same measure as this 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 a EB triangle a II B is similar similar is similar to triangle D E C triangle D II see 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 the ratio of corresponding sides are going to be the same so we could say that the ratio of let's say the ratio of B II the ratio of B II let me write this down this is this side right over here the ratio of B ii ii ii ii ii ii ii ii ii ii is going to be equal to so that side over that side well what is the corresponding side the corresponding side to be e is side c ii so that's going to be the same as the ratio between c e and de and de and this just comes out of similar the similarity of the triangles ce2 de so once again once we establish these triangle is similar we can say the ratio of corresponding sides are going to be the same now what is what is the ratio between be e and AE the ratio between B II and AE well that is the slope of this top line right over here we could say this the slope of line a B slope 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 a B 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 V E 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 see e over de c e 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 to 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