So what I'm giving here is I'm giving some information and I'm going to compose a two column proof. So I'm going to compose a bunch of statements on my left side and a bunch of reasons for those statements on my right side. And ultimately logically given all these statements that I'm trying to prove at the end of this fire to go all the way down to the end. My final statement should be that triangle abc is congruent to triangle. Um E. D. Now how do I how do I prove that to? Triangles are congruent? We'll end this lesson. I've talked about a couple of ways to prove concurrency right. I could prove that three sides are congruent and both triangles. I could prove that given two sides are congruent, an angle between those two sides um then two triangles would be equal or I could prove to his Bangles are equal in the side between those two angles. Given one of these three criteria, I'm ultimately trying to prove three sides. Two sides in the main goal to ingles or aside to prove that these two triangles are congruent. So let's see what we start with. Okay, what am I given first? I'm given that side A. B is congruent to side E. And this is already a side and the reason for this is given. Okay, I've already proven that one curve sides equal. Okay, what else am I giving? I'm giving that ray E. And A C tricep angle E. A D. What's the reason for this also given. So let's let's do a little bit more. I've been proven an angler side here. So I'm gonna use the statement of proven anglers side. Okay, E N A C tricep V A D triceps. What does that mean? Three insect means cut it cuts this angle into three equal pieces. So from this, I could say angle B E. Is equal to angle E. A. D. Which is congruent to angle E. Uh sorry E. A. C. And E. A. D. And what am I? What's my reason here? I'm just using the definition of tricep and an angle triceps. Okay. Does this prove that um this has proven anything. We'll know. What are my two angles? My two angles here. I'm trying to prove that angle abc is congruent to angle E. D. But I know that angle Abc is angle B. E. Plus angle E. A. D. And I also know that angle E. A. Um Sorry this is C. A. D. Up here. C. A. D. Okay, another angle E. A. D. Is angle B. E. Plus angle uh C. A. D. How did I know this? Well this is just angle edition. Right. I'm just I see in the in the figure that abc is separating the two angles and the baby is separated into two angles as well. Okay, So why is this important? Well, now what we're gonna do is we're gonna just subtract these two statements right? A. B c minus angle abc minus angle E A. D. And just attracting these two statements, I see that angle B a minus angle be a cancels and E A. D minus angle C. A. D. This will cancel because these two angles are equal. So here I'm just doing subtraction. And then my next statement is going to be addition to both sides, right? Which ultimately states that angle abc is congruent to angle E. A. D. Which is what I'm trying to prove. Okay, that's an angle. So I have a side up here. I have an angle up here. Okay, what else can I say? What are my other given statements? Another given statement is that A. B besides A B is perpendicular to side B. C. And I also know that E. Is perpendicular to do. This is also just given what can I stay from this? Right? I need another angle or aside from this information. So I'm going to harp on this little more. Okay, angle A B. E. Is perpendicular to angle B. C. So that means that angle B is 90 degrees and angle A is perpendicular to angle D. E. So the angle E is 90 degrees. And here I'm just using the definition of perpendicular. Okay, what's the point of this here? Well, this means that angle be is going to be congruent two angle E. Since they're both equal to 90 degrees, I can do transitive property or substitution. And then this is another angle. I see that given these two angles, the sides that were initially given our between those two angles. So finally, my final statement, these two triangles are equal by angle side angle concurrency from the roof above.