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00:39

Evan S.

Classify each statement as true or false. $A, B, C,$ and $D$ are coplanar.

01:05

Make a sketch showing four coplanar points such that three, but not four, of them are collinear.

00:37

Classify each statement as true or false. $R$ and $S$ contain $D$.

00:18

Classify each statement as true or false. $h$ is in $R$.

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Hey, guys, we're here to talk about similar triangles today, which, after a lengthy discussion about congruent triangles, which we're gonna review quickly before we move on to similar triangles, you're going to see, um what, Well, what's similar between triangles that are congruent triangles in a similar and the things that are special Two similar triangles. So let's review congruent triangles recall if we have a triangle and then another one, and though the exact same size and shape, I'm gonna call this first one ABC and the second one x y z, my markings are going to indicate which sides are congruent to each other and what angles are congruent to each other. Remember, if you are a congruent pair of triangles, you are the exact same size and shape, which means all of your respective sides and angles are congruent to each other. Also recalled that when you write, your congruent statement is very important about the order. So in this case, if I chose to say Triangle, ABC had to say triangle X y Z based upon three order of the sides that are growing in the angles that I grew up and recall it, the symbol for congruence e is basically an equal sign with, like, a tilde on top. So why is this important? What orders important So we can tell what's going on. I technically wouldn't even need the picture. I could say Well, based upon Maiken grew in statement. I know that a b is congruent to X y I would know that angle, see is congruent angles e based upon the congruent statement. Okay, so here you have congruent triangles, same size, same shape all sides and angles are equal in length and angle. Measure two. That's respective partners on the other triangle. What about similar triangles? Well, similar triangles are triangles that are the same shape but not necessarily the same size. So, for example, you could have this triangle here and you could have this triangle here and again. I'll call them ABC and X y Z and but this time there's gonna be a ratio of 1 to 2, which means all of these sides and the bigger triangle are double all the sides and the smaller triangle. So, for example, if a B were 10 units, X Y would be 20 units If XY were, let's say 40 units than a C would be 20 units. You have this wanted your relationship so whatever is you know whatever length of a side and triangle ABC all the length are doubled in triangles X y Z Interesting. The interesting thing is the angles, though. Don't follow that pattern. The angles air actually can grow into each other, respectively. So that's what they have in common in terms of congruent triangles. So we have the same shape triangle, but not necessarily the same size, which means there's a skill factor going on. So if you want to consider congruence, see triangles that are congruent a special case of similar triangles. In fact, they are where the proportionality, the scale factor is just one. There's a 1 to 1 ratio. So if you have a 1 to 1 ratio of similar triangles, there congruent because the sides are going to be in proportion to 1 to 1 like and growing triangles. It's important how we write the similarity statement. I will say Triangle ABC is similar. Notice we take away the equal sign part. We just have that tilda looking thing. Two triangle X y Z. Now, instead of saying a B is equal to x y in a similarity statement because we have proportional triangles instead of saying a B is congruent to x y. What I could say is stuff like this a b is two x y as BC is toe Y z. So now you can make these proportionality statements these ratios that that air can grow into each other because of the scale factor between one triangle to the other. You could technically do that to the other triangle that are congruent. But in that case, these would be equal and these would be equal. And so you basically get equals equals which is what we're saying income triangles, anyway, that one side equals the other. So this is very special to similar triangles where you can say we have one side is to the other as another side is to another a za, long as you're choosing the correct order. But I could also say, just like congruent triangles, I could say you know what angle B is congruent to angle? Why? Because the angles are equal respectively. No, just like in congruent triangles, we have side, side, side, side angle, side angle, angle side and angle side angle, which are four congruence therms, saying, If you have like for example, three sides of one triangle could grow into three sides of another triangle those side those triangles with the same size and shape, respectively. Well, with similarity, we have side side side. We have side angle side, but this time all you need is angle angle. If you have two angles equal in one triangle to two angles of another triangle, you have least similarity. So notice we don't say angle angle angle because if you have two angles equal to two other angles, it forces the third angles to be equal to each other biologic. So as opposed to four congruence terms, there are three similarity. The're ums. So if you let's say, have instead of three congruent sides of you know, three sides could grow into a three other sides of here, you have to have three proportional sides. So if you have three sides of one triangle proportional by the same scale factor to three sides of another triangle, they are similar. So let me show you, hear what I mean by this? Let's let's look at like the say for example, the side angle side similarity. So if I have this triangle and I have this triangle and let's go ahead and label this again, I know I'm being born with a B, C and X y Z, but I'm just trying to make it easy. All right, let's say I know that a B over X Y is equal to B. C. Over y Z knows my lettering is very important, but I also happen to know that angle B is equal to angle. Why? What I've shown you is these sides are proportional to these sides, and the angle that's include between them are equal. That's an example of siding with side similarity. You have two sets of proportional sides and the angle in between them. The angle included included angle. However you want to say it is equal. This is an example of how you would prove two triangles to be similar. Likewise, if you have to similar triangles, if you prove to triangles or similar, you can then say every set of sides is are proportional and every pair of related angles are congruent, so I'll be back in another video with some examples of how you might use similarity and algebra. I'll also be back with another video of how you would use formal proof to prove to triangles are similar Thanks.

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