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Right Triangle Congruence and Examples

In mathematics, the term congruence refers to the relationship between two objects, or sets of objects, when one of the objects is said to be "similar to" or "congruent to" the other. This relationship is denoted x ~ y, and the notation is read "x is congruent to y".


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Video Transcript

Hey, guys, we're here to talk about congruence. See with right triangles? No. In some previous videos, I did talk about how to prove triangles, congruent triangles in general. First of all, if you recall from grew and see means you have a shape that's the exact same size and shape where all the sides and angles respectively are equal to each other. So specifically with triangle he talked about side, side, side, side angle, side angle, side angle and angle angle side concurrency And I didn't mention Cem, right? Triangle congruence the're ums. So if you recall if you have two right triangles we had several therms that would prove the two triangles are congruent and those were high pot noose leg hi pot news angle, leg, leg and leg angle. These are the four high right triangle congruence firms. So essentially, if you have an iPod news and the leg congruent to one another than the triangles air or if you had high pot news and an angle or a leg in the leg and so forth. So as long as you meet any, you know, to requirements labeled here and you're gonna have the right triangles, conclude into one another. So here's some examples of that. First of all, some simple. Can you identify if two right triangles when grown or not? Remember going right over here to the side. We have high pot news leg hi pot news angle, leg, leg and leg angle. We're just gonna look through these these these examples and see if we can get any one of those to be true. So if you look at example or question one well, when I say by the way, leg angle or high pot news angle, we're not referring to the 90 degree angle. We're referring to the other two angles in the triangle. If you look at example one, there's no angles besides the writing labeled. But we do have that they share this side so we have leg leg congruence. See here because we have one leg congruent to the other because they share that one and then this leg and this leg, as indicated by the tick marks on question to you can see how we have hi partners and then a leg. This would be high pot news leg unquestioned, for I'm just gonna skip around. You can see how I have the pot news here. This is a right triangle. Because of vertical angles, we have one angle referenced. So here we have high pot news angle. Okay, so here's basically a zoo. Long as you can identify any one of those four situations than you can state that you have, uh, congruent right triangles. And you know, here's a few others examples or types of problems Where now I want you to prove to me that the triangles are congruent by the indicated congruence here. What would be the piece of information that we need in order to say yes, these triangles are congruent Well, this is high pot news leg. We have the hypotenuse apartment. We have the hypotenuse of each triangle already equal. So pick a leg. How about DF and VX? So we would have to have DF equal or congruent to Vieques notice My order is important because I'm starting at the right triangle and going to the next Vertex. I want to start at the right triangle and go to the next Vertex When questioned. 13 leg leg. We already share a leg, so we need to force the other leg to be congruent so I would need in this case h m to be congruent toe l What I needed to do, by the way, is put my segment bars on top. So this would be an example of Okay, Force The triangle is to be congruent via this. I via this congruence The're, um will be the missing piece. Okay. And then if I were to let's see what? Like how toe prove Two right triangles can grow. Let's suppose I have this situation apartment. Let's suppose I have given here my two givens angle A is congruent to angle See and angle B D c is right. What I'm asking to do is prove triangle a B D. Congrats to triangle. See BD Well, with the previous congruence teams, I wanna label what I have. Okay, First of all, I know that the sides could grow into itself I also know that by our givens those two angles were equal. This given BDC is right is basically telling us that that's the right angle, which means we have two right triangles. Okay, so if you're gonna do a proof with a right triangle congruence their first, you have to state that they're right. Triangles then state the two things you would need. Like hi pot news in an angle or partners in the leg and so forth. And the last statement would be your congruent statement and may be your last last statement is gonna be maybe something involving a C p c T c. Would you be congruent parts of congruent triangles or congruent? So in this case, what I would first do is I would say something like this angle B d. C is right. My reason would be given. Then I could say angle B. D. C is a right triangle. And I could say, Remember when in doubt, Julie Definition of definition of right triangle. Okay, now, because we're gonna assume this is a straight angle, we're gonna be really picky of to prove it to straight angle and prove that those back to back angles are both right. But this is basically going to imply that triangle, and I gotta make that a triangle. Simple triangle. If we said BDC, I'm gonna say triangle B d A is also right. Kind of like an implied Okay, so we have right triangles. Okay, Now I can say that face from my picture. Okay, that b d is congruent to itself. And remember that when you talk about something being group to itself, that property is called reflects of property. Now I know that also angle A is congruent to angle. See, that's given. And then what I can say is triangle A B D is congruent two triangle CBD. Let's talk about what we have here. We have an angle and we have basically a leg. This would be because of leg, leg angle concurrency, which is one of our four right track walking around storms. So just like the yeah, other problems, the other proof problems that we saw earlier You have to establish what you have, the givens or the picture. Then use those things, obtain your three or, in this case, to required pieces of info. They're going to get you to one of your in this case for right triangle congruence, Ethier ums. And then once you have that established, you could do your congruent statement. Order is very important and let's say I had said something like instead of prove a, B, D, e and CBD or congruent, I might have said prove, You know a B is congruent two c b Then my last statement would have been a B is congruent two c b and that would have been by C P, C T C and you remember what that means. It stands for a congruent parts of congruent, triangles congruent Or you might also say just definition of congruent triangles. So either one will work CPC TCs like kind of ah, well known well used kind of acronym. But you could also say definition of congruent triangles. Okay, so whatever you need to do to get to you're congruent statement. Okay? Again, you're gonna need to pieces and notice like this Waas my first piece. And this is my second piece because all of our right triangle congruence terms require to pieces. Um, so use what you have to get to what you need, which allows you to make your congruent statement and if necessary, go to the last step to prove their meaning. Pieces are congruent to each other. We could have gone even so far as to say, prove that BDs and angle by sector of angle, ABC We draw this picture again. I could have said proved that prove it. B d is an angle by sector of angle A be see, meaning those English or equal. We know this angler equal because we proved the two triangles can grow. We eventually prove that this triangle is congruent to this triangle by leg angle. And you would say that those angles are equal by a CBC TC, and then what you could say After that you could go one step further and say BD must be an angle by sector. Because if those two angles are equal and that's bisecting it, well, by definition is an angle by sector, so you could always keep going further and what I want you to think about. What you're doing approve, is what would I need to get there and work backwards? Don't you think about your end result? What would you need? What you need? And typically, once you go back far enough, you could say All right, here's my information I have This gives me the three of the two pieces that I need to prove triangles congruent. Once I had the triangles congruent, I need to be done. That's all may be asked to do. Or I could do the next step and say these two pieces or angles or so forth are growing to each other or can go even further further and prove something that those congruent pieces imply. For example, two angles being equal. Maybe I have an angle by sector two segments being equal. Maybe I have a midpoint that I've proven exists something along the lines of that. So and here This is one example of a proof with right triangle congruence. And previously we saw just a couple of examples of how to verify whether or not to write channels that grew up or how to find the missing information in order to ensure the two right triangles are congruent by a given right triangle theory. Okay, I'll see you next time. Thanks.