Topic C: Congruence and angle relationships
Congruence and similarity — Basic example
- [Instructor] In the diagram at left, I'd wrote up here above or I pasted it up here, line segment AD intersects BE, and line segment AB, AB is parallel to DE. What is the length of side CE? So we wanna figure out the length of side, we wanna figure out the length of side CE. Now it might jump out at you, at least it looks like it, immediately that triangle, the small triangle on the left is going to be similar to this larger triangle on the right. But let's feel good about that, let's prove that to ourselves. And to help ourselves prove that we're gonna remind ourselves that AB is parallel to DE. And actually to think about that even more, I'm gonna, I'm gonna make, I'm gonna extend these segments a little bit so that you can really start to see that they are parallel lines, and that we have a transversal, or we have two transversals that intersect those parallel lines. All right, and I could keep if I wanted to extend this as a line, I could keep going in both directions. But the bottom line, not to be funny about it, is that these two things are parallel. That is parallel to that, and that's going to help us establish that these two triangles are, that these two triangles are similar. Well the first thing we know is look this angle, is vertical, is a vertical angle with this angle, so they're going to be congruent. This angle right over here, if you view segment BE as a transversal of the two parallel lines, that, that right over there is going to be an alternate interior angle, this is going to be an alternate interior angle to this angle right over here. These are alternate interior, alternate interior angles, so they're going to be congruent. If some of this sounds like Greek to you and you are not Greek, I encourage you to watch the videos on Khan Academy of angles, of transversals, and parallel lines. And by the same argument, if we look at AD as a transversal, this angle right over, well let me do that in a different color, this angle, this angle right over here, is an alternate interior angle with this one, so they're going to be congruent. But the whole point of everything I just did is to say look, the three angles here are congruent to the three angles here, so these must be similar triangles. And the reason why that's useful, is because if they two triangles are similar then the ratio between corresponding sides is going to be the same. So for example, the ratio of side CE, so we could say the ratio, the ratio of CE, of the length of segment CE to say, to say this side right over here, so side segment ED, which has length 7.5, is going to be equal to the ratio of the corresponding side to CE. Here look CE is a side opposite this green angle, so if you go on this similar triangle, opposite to this green angle, that's segment BC. So it's going to be equal to the ratio of the corresponding sides, so 2.1, and the other triangle to the corresponding side of DE in the other triangle. And this side the opposite this blue angle, this, this, one of the vertical angels right over here. So the corresponding side over here is going to be side BA. So this is going to be over 2.5. So what would CE be equal to? Well multiple ways to solve this, but one way we could just multiply both sides by 7.5, multiply both sides by 7.5. Now what's 7.5 divided by 2.5? Well 75 divided by 25 would be three, so 7.5 divided by 2.5 is three. So this is going to be three, so length of segment CE is 2.1 times three. So the length of segment CE is going to be, what is that 6.3. And if we look down here, we see that that is indeed, that is indeed one of the choices.