 In this video we're going to talk about the angle addition postulate in geometry. Basically we're just going to go over just a few examples of angle addition postulate. It's a rather simple postulate. It's kind of self-explanatory. Angle addition means you just take the angles, add them together to get a larger angle. It's basically what it boils down to the simple version of it. I'm not going to go over the definition of angle addition postulate. I'm just going to go over a couple of examples that we're going to have. Anyway, this first example that we're going to look at. Find the measure. Remember this little M here means measure. Find the measure of angle FEG if the measure of angle DEG is equal to 115 degrees and the measure of angle DEF is equal to 48 degrees. One of the first things that I'm going to do since they actually tell me a few things is I'm going to label some of this on this little drawing that they give me. They tell me I'm supposed to find the measure of angle FEG. If I look here, FEG, that's this smaller one down here. Right here I'm going to mark that little arc right there. That's the angle that I'm looking for. They tell me a couple of additional things. If the measure of DEG, which is here, DEG, we're talking about the entire angle. This big one here, this one is 115 degrees. I'm going to use a big arc right here to denote that the entire angle is 115 degrees. The thing is I only want to find a little piece of it. The other piece of information that they tell me is the measure of angle DEF, which is DEF, is equal to 48 degrees. I'm going to put a little 48 right here to help me with that. Angle addition postulate tells me if I take this small angle plus this small angle that they should add up to 115 degrees. I'm actually going to write that out. It is a little bit lengthy, but I'm going to write that out so we see the notation. The measure of angle DEF plus the measure of angle, let's see, DEF and then FEG. We'll call it measure of angle FEG is equal to, if I take these two angles, this angle here, this angle here, if I add them together, I should get the larger angle, which is angle DEG, measure of angle DEG going over to my other problem here. Anyway, so now that I have, now that I have, this is my formula. So in geometry, we're using formulas every once in a while. It's my little formula. Take, here's a small angle plus the other small angle that's going to be equal to the larger angle there. So now what I'm going to do is I'm going to take some of the information that I know from my drawing and put it into my little formula here. So DEF, I actually know what that is, that's going to be 48 degrees plus the measure of angle FEG, FEG. I actually don't know what that is. That's actually what we're looking for. That's what they tell us in the problem up here. So measure of angle FEG, I'm just going to leave that, and then equals the measure of angle DEG. That is something else they tell us, which is 115 degrees. So 115 degrees. So now we can use just a little bit of algebra to solve this. I'm going to treat this right here. I'm going to treat it just like a variable. You can even put in a variable there if you want to. So 48 plus x is equal to 115. So small angle plus another small angle is equal to 115 degrees. We just got to find out what that other small angle is. So all I got to do is just subtract 48 from both sides. x in this case is going to be equal to, what is that? 2, let's see, 15, 67. Hope that's right. 67. You can also check to see if you did that right. What if I take 67, if I plug it back up into here, the two smaller angles are 48 and 67. So if I take 67 and 48, and if I add them together, I've got a little extra going on there. 48 and 67, that's a 5 right there. That's 10, 11, which is 115. Yep. So just with a little bit of quick addition, I'm able to figure out that I did do that correctly. Okay. So in this case, this right here, that's not my answer. I have no idea what x is. We've got to actually answer the question, find the measure of angle F, E, G. So the measure of angle F, E, G is equal to 67. Degrees, make sure you get your labels on there. Okay. So there we go. And this is, again, using the angle addition postulate to solve this. All right. So that is one problem. I'm actually going to have to erase a little bit of that because it went over into my other problem here. Okay. So let me erase a little bit of this. Let me erase this here. Get rid of it. Get rid of it. All righty. And again, you can always rewind this video and go back and see what we previously did if you really, really want to. So I'm not too worried about it. All right. So the next problem that we have here, find the measure, find the measure of angle PQS if QS, the ray, bisects angle PQR. And we also know the measure of PQS is, okay, expression 5y minus 1 degrees. And the measure of PQR is equal to this expression, ay plus self. There's a lot of information here. And there's no picture. Okay. So that's one thing we got to do. We got to come up with a picture ourselves. Okay. So now the first thing that they say is, find the measure of angle PQS. You know what? I'm just going to draw that angle. It doesn't really matter where I draw it or how big or small I draw it, but I'm going to draw it so I have something to work with. Okay. So here's P right here. Here's Q over here. And there's S right there. Okay. Little arrows to help me out. Little arrows to help me out. Okay. There's my angle PQS. Now, again, I know PQS, everything goes in order. P here. Q is the vertex in the middle. And then S is on this side here. And again, actually we could switch around P and S, but that's, we don't necessarily need to do that. Okay. So that's the first part for drawing my picture. So that's the first part. If ray QS, okay, if ray QS bisects PQR. Okay. So now we've got another angle here. QS is a ray. Notice here I have QS. There's a ray right there, which is just part of the angle. It's just a ray. Okay. That already exists. And it bisects, it cuts in half this other angle, angle PQR. Well, PQR, I haven't drawn that. Where is that? Okay. I'm going to draw myself another angle. Okay. Now this is where working with pencil might be a little bit easier, because now what I'm going to do is I'm going to draw angle PQR. But PQR has to be cut in half by ray QS. So actually it would be somewhere over here with my current drawing. QS cuts PQR in half. Now this is actually, this is a pretty bad drawing. I'm going to admit, this is a pretty bad drawing. This doesn't really work, because PQR, how I have this drawn, it looks like a straight line. So you know what? I'm going to redo this. Okay. And this is okay to do. You're drawing your own picture here. You're going to make mistakes. I made a mistake. No big deal. So what I'm going to do is I'm going to redraw this. But now I have the knowledge that QS is going to cut in half PQR. So this is a little bit better drawing. Notice that QS is right there in the middle. It takes PQR and it cuts it right in half. Now one thing that we know since it cuts it right in half, this angle right here and that angle right there, they're congruent. These two angles are going to be the same measure. Okay. We're probably going to use that here in just a moment. Okay. So I got a lot of colors. I'm going to change up my colors here just a little bit. All right. Moving on with the rest of the problem. The measure of angle PQS, measure of angle PQS is equal to 5y minus 1. Now over here with this other problem, we had just numbers. Now we're getting expressions. That's not a big deal. We're just going to write them down just like we did last time. So measure of angle PQS, PQS, which is this top angle right here, is 5y minus 1. Okay. Now I'm going to forego the parentheses and the degrees. I'm going to assume that we know that this is going to be in degrees. Okay. All right. So instead of a number, we're going to do a variable expression there. Okay. No big deal. And measure of angle PQR, PQR, which is actually all of this. It's the entire big angle. Okay. It is going to be 8y, 8y plus 12. Okay. So now that we've labeled everything, now we've got to try to figure out what the measure of PQS is. So PQS, that's this small angle right here. That's actually what we're looking for. Okay. Well, we can't say, oh, it's just 5y minus 1. No, not really. We want the measure of it. Okay. We're looking for an angle measurement. So we need a degrees. Okay. So what do we do? What do we do with all of this? Okay. Well, one thing that we do know is, if we have these arcs here, we know this angle is 5y minus 1. That means this angle down here is also going to be a 5, 5y minus 1. Okay. So those two angles are actually the same. Yes, we're not using numbers here, but we can still use variable expressions. It's not a big deal. Okay. Now the thing is, we also know that these two angles with the angle addition postulate, these two angles, if you add them together, you should get the entire big angle of 8y plus 12. This gives us the ability to actually write out an equation for us to solve. Okay. Not a very complicated one. So if I take 5y minus 1 plus 5y minus 1, so I take these two angles, if I add them together, I should get 8y plus 12. There we go. Now notice I didn't write a measurements and angles and all that kind of stuff like I did with this last problem, and that's okay. I'm just kind of foregoing that, and I'm just doing the algebra right here. Okay. But again, it's very similar to the last problem. Okay. So now we're going to solve this. So what do we need to do? We need to get everything on one side, variables on one side, numbers on the other. Notice here I see a 5y and a 5y. That's going to be a 10y all together. Negative 1, negative 1 makes negative 2, and then over here, 8y plus 12. Okay. So now what I'm going to do, I'm going to do two steps at once. Since I'm getting to the bottom of my problem here, I'm going to subtract the 8y over here to make 2y, and I'm going to take this 2 and add it over to the other side to get 14. That means, getting to the bottom here, y is equal to 7. Woo! No. Wait a second. A lot of students will make this mistake. I don't care what y is. I really don't. I want to find the measure of angle pqs. This is going to help me. That's not my answer. Okay. That's not my answer. That's going to help me find my answer, though. So y is equal to 7. What I'm going to do is I'm going to take this and I'll plug this back into what I'm looking for. pqs. That's what I'm looking for. pqs is what I'm looking for. So take this 7 and plug it back into pqs right here. So that 5y minus 1 put a 7 in there. Let's see. 5 times 7 is 35. Minus 1 is 34. So that's what this 7 is used for. So the measure of angle pqs is equal to right here 34. Don't forget your degree symbol. Make sure you actually answer the question. Don't give me this. Don't give me y equals 7. Don't just give me a random number. Actually answer the question. Go back up to the original problem if you have to. Okay. Those are two examples using the angle addition postulate. Hopefully that helps you. Hopefully that gives you a couple of tricks and a couple of strategies to use to solve geometry problems using angle addition postulate.