Proof: parallel lines have the same slope | High School Math | Khan Academy

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.

Now I'm going to draw some transversals here. So first, let me draw a horizontal transversal, just like that. Then let me do a vertical transversal, so just like that. 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.

From this, I'm going to figure out—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 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.

Let's see. First of all, we know that angle C D is going to be congruent to angle A E 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 a transversal where a 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 point B is going to also be congruent to that because they are vertical angles, and we've seen that multiple times before. So we know that this angle, angle A B, is congruent to angle E C D. Sometimes this is called alternate interior angles of a transversal and parallel lines.

Well, if you look at Triangle C D and Triangle A B, 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. This third angle is just going to be 180 minus these other two. So just like that, we notice we have all three angles are the same in both of these triangles—or 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 A E B is similar to triangle D E C by angle-angle similarity. All the corresponding angles are congruent, so we are dealing with similar triangles.

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 E to A E is going to be equal to the ratio between C E to D E. This just comes out of the similarity of the triangles C to D E.

Once again, once we establish these triangles are similar, we can say the ratio of corresponding sides are going to be the same. Now, what is the ratio between B E and A E? The ratio between B E and A E, well, that is the slope of this top line right over here.

We could say that's the 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 A E and your change in Y is B E, 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, C over D E. Well, this is going to be change in Y over change in X between points C and D.

So this is a slope of line C D. 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.