 Okay, and welcome back today. We're going to talk about two things We're going to talk about the hinge theorem, and then we're also going to talk about the converse of the hinge theorem now Of course just by their names you can tell that they're going to be very very similar So I kind of doing this all in one video. All right, so let's start off with the first one Now whenever you read these theorems and mathematics make sure you I'm going to read this all through once But when you read these theorems a lot of times you cannot figure out what is going on when you read all the way through them So I'm going to read through it once and then we're going to go back and kind of explain it piece by piece. All right Here we go if two sides of one triangle are congruent to two sides of another triangle and The included angles are not congruent Then the longer third side is a cross from the larger included angle Get now obviously again when you read all the way through that it's a little bit confusing But let's get through this piece by piece make it a little bit easier to understand. All right So if two sides of one triangle are Congruent to two sides of another triangle. All right, so you notice my pictures down here So two two sides of one triangle congruent two sides of another triangle So right here one tick mark one tick mark. There's one pair of congruent sides Two tick marks here and then down here two tick marks here There's my second pair of congruent sides, okay? So that's what that first part that's first part of that sentence talking about okay, and the included angles are not congruent okay now included angles simply just means the angle that is included in Between the two sides that we're talking about okay, so we're talking about these two sides here So the included angle is gonna be this 50 degrees here same thing over here on this triangle Two sides. We're talking about these tick marks right here The included angle is going to be this angle right here this 28 degrees All right, and they are not congruent so they're not the same obviously you can see that this one's 50 degrees This one's 28 degrees. They're going to be a little bit different all right now last part then the longer third side is Across from the larger included angle all right So basically what it means the longer third side is going to be across from the larger included angle So basically what that means and it's kind of you might be already see this here's 50 degrees Here's 28 degrees 50 degrees is the larger angle 50 degrees is the larger angle, which means that's a horrible arrow there Okay, which means this 50 degrees means that this AC is going to be larger 28 is smaller 28 is smaller. So then that means That this is going to be a small side. So then what do we know? We know that this side AC is Going to be greater than this side over here x z Okay, so basically what this hinge theorem tells us is this is a way for us to compare sides of Triangles from two different triangles. So again, we were comparing this third side AC over here with this triangle's third side Xz over here, but again, we had to have some we had to have some criteria first That's why we talked about at the very beginning. We talked about here's a current ruin side Here's a congruent side same with another pair down here You got to have that first and the included angle. So there's a couple of things we have to have Before we can actually start comparing those two sides. Anyway, there's the hinge theorem Okay, now on to its counterpart to the converse of the hinge theorem Okay, now instead of reading this all the way through like I did last time for the first one I'm just going to go bit by bit Kind of explain our way through this. This is very very similar. So I'm gonna go a little faster Okay, if two sides of one triangle are congruent to two sides of another triangle So again, we got our tick marks again. So here's one pair of congruent sides Here's another pair of congruent sides. There we go And the third sides are not congruent. So notice here. We got this 13 and we got this 18 Those are obvious that the third sides of these triangles are not congruent. They're not the same Okay, so now that we've met that criteria then the larger included angle is a cross from the longer Third side. Okay, so what we're doing here is we're looking for the larger angle So this 18 over here is our larger side, which makes the opposite angle larger This 13 is the smaller side. It's gonna make this angle over here smaller Okay, so then and then what do we know down here now? This could be a little bit different from the first one We're not looking at sides this time instead. We're looking at comparing angles. So actually Well, we'll start with this angle over here The measure of angle C is actually going to be smaller than the measure of angle Z Okay, Z is actually the bigger angle. So I use my less than symbol there to compare those two angles Okay, all right. So that's the hinge theorem and the converse of the hinge theorem is going over those rather quickly All right now what I'm gonna do is I'm gonna go over a couple of examples that uses these theorems Okay Now the first thing we got to do whenever you're comparing the measurement of angles like we're doing for this first problem Or if you're comparing the length of sides like we're doing the second example The first thing you got to do you got to recognize your triangles. Where are they? You also got to recognize make sure you have those two pairs of congruent sides Okay, so the first thing I'm going to do when I look at these when I look at my picture is I got to find my pairs of Congruent sides and then I can start comparing these angles we have up here. All right, so now when I look at the picture I got two triangles I got this one triangle on top and I got this second triangle on the bottom and I'm going to look for my I'm gonna look for my congruent sides. Now the first obvious one is right here five and five There's my first pair of congruent sides My second pair a little bit harder to see is actually this pair right there This pair right here in the middle is the bottom side for the top triangle This side is also the top of the bottom triangle Okay, there's that that that right there is actually two sides It's it's one for the top triangle and one for the bottom triangle. So we got to kind of count of twice So there's our second pair of congruent sides. All right, so now that we have that now We can start comparing the angles. Okay. Now what I'm gonna do is I'm gonna color code these a little bit We'll make MLN MLN right here is gonna be our red angle. All right, and then PLN the other one. We're gonna compare with PLN is Going to be right here Okay, so now that I have those marked up now I can instead of saying MLN and PLN I'm just gonna say the red angle the blue angle makes it a little bit easier for us Okay, so now what I'm gonna do is I'm gonna compare them So now as I look at these six is going to be the small side Making red my small angle seven is the big side Seven is the big side. You say larger or big. Okay, so that makes blue my bigger side So what that means is that the measure of angle PLN Is larger than the measure of angle? MLN there we go. Okay, so what we were doing is we're using these these third sides over here the six and the seven We're using those to help us to defeat to figure out which angle is going to be bigger Which angle is going to be smaller? Okay, all right now down to the second example down here now What we're doing is we're comparing sides, so I'm going to do kind of the same thing I did last time what I'm gonna do is I'm gonna color code this first Okay, so XY is this side. We're talking about right here trying to make a straight line there Okay, ZY is this side over here. All right Now as you look at this problem The first reaction you see is that well if I'm comparing the length of these two sides I know which side is bigger. Okay, this side over here XY is obviously a lot bigger And yes, you are correct in that but the thing is is in mathematics what we got to do is we have to prove everything You have to have evidence of what you're thinking you can't just say oh that side is bigger because it looks bigger That doesn't really work anymore. What you have to do is you actually have to find evidence of what you're thinking of Okay, so what we're going to do is we're gonna identify our triangles just just like what we did last time with this example up here Okay, so I have two triangles I got my one on the left here, and I got the one on the right here And I got to find the pairs of congruent sides, so here's a 12 and a 12 There's one pair and then just like last time up here The this side is shared between both the triangles, so there's my second pair of congruent sides This is the this is a side for the left triangle, and this is also a side for the right triangle Now that I have those pairs of sides now what I can do is I can start looking at the angles Well notice that we're actually this 137 that helps to make this side over here really big 137 is a pretty big angle, but the thing is is we're missing We're missing the angle right here to help us identify with this side Okay, so now what we have to do is go back to some of our previous knowledge This right here is a straight line straight lines are a hundred and eighty degrees So what I what I can do is just take a hundred and eighty and subtract that from a hundred and thirty seven or excuse me other way around 180 minus 137, and I think I get forty three degrees 43 yep, that's what that is so now that we have an angle measurement there now We have for sure we are hundred percent sure that this yz is going to be the smaller side and then this 137 makes xy the larger side, so now we know that xy is going to be larger Than zy okay, so that gives us the ability to write that again. You got to have evidence of it You can't just say willy nilly I think this side is bigger than that side because I think so you can't really say that you got to have some evidence to that already that's it for The hinge theorem and the converse the hinge theorem. Thank you for watching this video, and we'll see you next time