Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
When a line is drawn parallel to one side in a triangle, two similar triangles are formed because corresponding angles yield the AA similarity shortcut. Because the triangles are similar, the segments formed by the parallel line are proportional segments. When finding one of the bases of the triangles, be careful in setting up the proportion since the ratio is equal to the small triangle's side to the large triangle's.
If we have a triangle and if I draw a line that's parallel to one of the bases, question that I'm going to pause is does that create 2 similar triangles?
Well, to do this we're going to have to say, one of our shortcuts angle angle, side angle side or side side side will have to apply in order for us to say that this smaller triangle dve is similar to the larger triangle abc. And notice that I've marked our angles 1, 2, 3, and 4. The reason why I did that is because I'm going to say that angles 1 and 2 are corresponding angles, which means that they must be congruent to each other. Since we have a transversal which is ab and 2 parallel lines, 1 and 2 are corresponding angles. In a similar argument bc is a transversal where we have 2 parallel lines which means angles 3 and 4 must be congruent to each other. And right now we have 2 angles in each of these triangles which is enough to say that they must be similar. So is triangle abc similar to triangle dbe? Yes, and our shortcut was angle angle.
So a couple interesting things happened here is we can use the converse of this and say that if you have 2 lines and the question is if this line is parallel, then you can say that these 2 triangles must be similar. And another way of saying that if these 2 angles are congruent and if these 2 angles are congruent then you must have parallel lines and you must have 2 similar triangles.
Let's look at 2 short examples. Right here I have a triangle and I'm being asked are we do we have 2 similar triangles? Well, if I look at this we have 70 degrees, 70 degrees, so those are congruent and we have 2 other congruent angles which means we can use the angle angle shortcut to say that these two must be congruent.
Now let's look at one other example. Here we have a triangle and again we don't have anything that's marked parallel. So what I'm going to do is I'm going to redraw my smaller triangle here. So this is the triangle with the side 4 and the side 6. Now I'm going to set up a proportion between the corresponding sides here. So we have 4 is the side on the left of the smaller triangle and the larger side is not 8 but it's 12 because the whole length is 12. And then over here we have 6 and the whole side is 18, 6 plus 12. So if I reduce these 4 twelves I can divide those both by 4 and I can get one third and here I can divide both of these numbers by 6 and I get one third. So are these triangles similar and the answer is yes. And our shortcut here would be the side angle side shortcut because they both share the same angle right there so it has to be congruent to itself.
Now one other interesting thing that you should notice is not only are 4 and 12 proportional and 6 and 18 proportional, but if I just looked at 4 and 8. 4:8 if I write that ratio here is equal to the ratio of 6:12. So if you have parallel lines, or if you have one line that's parallel to the base, you will create segments that are proportional to each other. So you don't even need to think about the ratio of 4 to the whole side. You could just say if we didn't know that length right there, that if this is 6:12 then this has to be 4 to some number. And so instead of using 8 we would have x and we would see that our ratio is double. So to get from 4 to x we would have to multiply by 2 and then we can find 8.
So two key things happened with a parallel line and a triangle. First key thing is it will create two similar triangles and the ratio of these sides created by that parallel line will be similar.