Equivalent ratios in similar shapes | Transformational geometry | Grade 8 (TX) | Khan Academy

We're told that quadrilateral ABCD is similar to quadrilateral STUV. So what we're going to do in this video, this isn't a question; this is just a statement right over here. But what we're going to do is think about what does similarity mean? What does it allow us to figure out?

Well, the first thing that similarity tells us is that the measure of corresponding angles is going to be the same, and they've actually already marked it up that way with the number lines. For example, we have one line right over here at angle ADC, and we can see the way they've marked it up that that's corresponding to angle UVS right over here. By putting one line right over there, they're saying that the angle measures are going to be the same. We see that for this angle right over here; they put two lines. That angle right over here corresponds to this angle, so they have the same measure. And I could do that for the other two.

So that's one thing that similarity tells us, but the other thing that similarity tells us is that the ratio of corresponding sides is going to be the same. Now, this is interesting because it could be corresponding sides across shapes or it could be corresponding sides within the shape. So what do I mean by that?

So let's first think about corresponding sides within each shape. So let's say we have side CD. Side CD corresponds to side or segment UV right over here. Notice it's going through the connecting the same two vertices. And let's compare that to, so let's think about segment DA right over here, which corresponds to segment VS right over here.

So one thing that we could say—and this is correspond—this is looking at the ratio of sides within the same shape—we could say that the length of segment CD, so the length of segment CD, if I write it like this, that means I'm referring to the segment. If I don't write that line, that means I'm referring to its length.

So this should be written as the length of segment CD. The ratio between that and the length of segment DA, then the ratio of the corresponding sides in the other shape to each other should be the same. So that means that the ratio—because we know they're similar—I can only say this because these are similar quadrilaterals. So I can only say that because of similarity. This is going to be equal to the ratio of UV to VS, once again, because those are corresponding sides.

Now we could also take it the other way. We could say that the ratio between corresponding sides is the same as well. So, for example, we could say that the length of segment CD, the ratio of that and the length of segment UV, so the ratio of this to this is going to be equivalent to the ratio of this to this of the length of segment DA to the length of segment VS.

And you have to make sure you're getting in the right order over here. I put from—I put the CD first, so I have to put DA first because they are from the same quadrilateral, because now I'm looking at the corresponding sides across quadrilaterals. If I were to flip one of these, then I would have to flip the other one of them. But obviously, there's many other things we could do. We could look at the other sides, but this is a really important thing to understand of what you can infer from the fact that these quadrilaterals are similar to each other.