### Proving Triangles Congruent Proofs

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### Video Transcript

All right. So here we are, the real deal. We're gonna prove to triangles can grow typically when you have a triangle and grow and stare, Um, you have a set of givens things that you know are true because they're given to you to be true. Then there's also these things that are true but implied True. So, for example, in this particular picture, I didn't have to tell you, but B d is equal to itself. So that's another given that's not directly stated. But it's there. So we have the blatant givens here and here, the indirect givens, which I just mentioned. And then you have this proof statement. We want to try to prove that triangles, Adeeb and CDB are congruent your last statement. Better be using or one of your last statements. Better be using one of the four congruence terms that we know side side, side side angle, side angle, ingleside or angle side angle. That must be one of your last statements, if not the last in this case, the last statement because the reason for saying that triangles are congruent bet every one of those four things so you could do this in a column proof a two column proof, which is traditional, you could do in a flow chart proof, which is what I prefer. And so I'm gonna try to show you both. There's gonna be a lot of vocabulary and statements that I'm gonna make during this that have not necessarily been in any of my videos, but hopefully you could pick up along the way in other texts, examples or classes that you take because to really be a good proof writer, especially if congruent triangle proofs. It does take a lot of practice. It does take a lot of, um, being there, uh, in the moment kind of thing. And so I'm gonna try to do my best to show you how to prove to triangles congruent while kind of like bringing in some terminology and things like that. Okay, so, anything to make a statement, you have to give a reason for that statement. Nothing can be unreasoned. There has to be a reason attached to it. And so here's what we typically do. If we're gonna be like a quote unquote traditional proof, I might say something like this. Step one angle A is congruent to angle, see? And the reason for that is gonna be given. And I'm just gonna put off to the side of the reason and simply it's given. You know what? Because he told me Waas, and I'm gonna mark the picture accordingly. It's very important to mark the picture super duper important. Then I'm gonna write statement to angle A B d is congruent thio angle. See, Be de well a BDs here I'm gonna put to tick marks. There are two cross marks so that I could differentiate between the other ones in CBD. What's the reason for that? Well, that's also given now. I mentioned earlier that we have B d congregant to itself That should be yourself, an obvious kind of dust statement and the reason that anything has ever congruent to itself. Okay, The reason for that is always known as this thing called the reflexive property. Okay, it is a very common reason in Jama tree anything that something is equal to itself. Reflexive property. Now I want you to see that we have an angle, an angle, and then aside, and then order an angle and angle. And then aside, that's one of our congruent storms. So what I can do right now is now that I have these three pieces of info pardon me. Now that I have these three pieces of info with their appropriate reasons, I can say that my fourth step is gonna be triangle A D B is congruent to triangle. See db. And what's the reason? Angle, angle, side congruence. That's your first proof. There you go. Four steps to direct givens. One indirect given because of a shared side. And then the three things came together to prove the triangles congruent. So this is how you would prove a very basic proof of how you would prove to tracks can grow. So ultimately, your last step or one of your last steps and you'll see why it might not be your final step needs to have either S s s s a s s or s a Okay, let's do another example. Okay, so here we're giving two triangles. Here's what we're given or given that B, C and D C or congruent, as marked with the one tick mark were given a C and E c or congruent as marked with the to tick marks So those are the two blatant things that were given. Remember, though we need three things to prove triangles from growing. And if I'm not giving you three things and either something I told you is going to lead to a third thing or the picture itself is going to give us something indirectly without having to state it. And you might notice that this angle here has to be equal to this angle there because there are vertical angles so I could go right to saying All right, first step BC is congruent to D c Reason given and for that matter, my second step is gonna be a C is congruent thio Easy again, given my third step is going to say angle notice. I'm going to say B c. A is congruent to angle. Now if I say his is important if I say b c A I have to say d c E and the reason for that is vertical angles. Okay, race that and do a better job. So we know that vertical angles are equal, so you could say vertical angles are couldn't grow it. And that's one of those statements you could just use from now on. That's true. Vertical angles or congruent. We've We've We've already accepted that. We've already proved that previously. So guess what now, guys, we have to go and race this we now have. This is very important. A side, an angle. And then aside aside, same angle and a side side angle side so I can go ahead and say All right, My triangles, ABC and E D. C Erkan grew it because of side angle side concurrency. And now we've just proven those two triangles can grow up. Okay, Okay. So you can kind of see in this picture where we're going with a statement, Because if we're given an angle here, were given a side here. And of course, these angles are equal because their vertical. Then we know that this is gonna be an angle side angle concurrency problem. So you can kind of see where this is going to go. You're gonna have one statement where you say this is true. Given a second statement where you say this given Ah, third statement where you say these two angles are equal. Why? Vertical angles are equal. Just like in the previous problem. The fourth statement is going to be that the triangles are congruent. There's gonna be 1/5 statement here. You see this? Okay, it should make sense that once I prove the triangles congruent that all of the remaining parts that we didn't know about should also be congruent. So you're gonna again First step is gonna be and second step are gonna be givens. Your third step is going to be this angle, and it's gonna because it's a vertical angle. Your fourth step is going to be saying one triangle is congruent to the other triangle because in this case, it's angle side angle. As we mentioned, the important new part to this one, though, is saying at the very end that ese is congruent to a C. In your reason, this is a new one. Your reason C p c t c. What the heck is that? Well, the stands for congruent parts of congruent triangles, air congruent. Let me say that again congruent parts of congruent triangles are congruent any time you're gonna tell me that one piece of a triangle is can grow into the other After you've proven the triangles could ruin themselves. You can use CPC TC Now some people like to say definition of congruent triangles because if you have two triangles are congruent, want all of their respective part speaking fluent? The answer is yes. So you can either say C P, c T c or you can say definition of congruent triangles. Whatever you like is acceptable. So they're interchangeable. They're the same. So here's what you're gonna have to do with your givens. Direct or indirect. Givens. You're gonna have to eventually come to a point where you have three things that point to your triangles being congruent. Those three things being either side side, side, side angle, side angle, side angle or angling aside once you have, that might be done because the proof might just say to prove triangles can grow and you're done. But if the proof says one step further and asked you to prove some subset of pieces congruent, you're gonna have to do that final step in either Say your CPC TC or definition of congruent triangles, and we're seeing a ton of, you know, like in this problem, to see how these ciders thistles shared side. So already I see a side here and I see a side here. Man would be nice if I had this angle in this angle, because then I would have side angle side. Well, look at these arrows. That means we have parallel lines. And if you remember from a previous video parallel lines and imply corresponding angles, So check this out. I'm going to say in my first statement, Yeah, that w x is parallel toe y z that's given and what's a direct result from that? Well, I could say that angle W x z is congruent to angle. Why z x and what's the reason? Parallel lines imply congruent corresponding angles. Okay, We also then have Z X is congruent to itself. We've already learned that's reflective. Therefore, our fourth step can be that the triangles or congruent why side angle side. Oh, you know what I forgot. Oh, my gosh. I forgot to say this, So this would need to be E could put this a step zero. Okay, step zero. You would say that these sides are congruent. That would also be a given because I said you have three things, right? And so what are my three things? The fact that the sides are congruent the fact that these angles or congruent, and the fact that the sites could grew into itself. This part about the lines being parallel doesn't say anything about a side on angle directly. What it does, though, is allow me to then say the angles or congruent, which gets me my one to and third piece of info for me, then get my citing besides situation. So here are a few examples of how to prove triangles congruent. I'll be back with some more.