 In this video, I'm going to talk about finding angle measures. So what I'm going to do in this video is go over a couple of examples of how to find angle measurements using the couple of different theorems and postulates that we've gone over, specifically the theorems and postulates over corresponding angles, alternate interior angles, alternate exterior angles, and also same side interior angles, all those postulates and theorems that I just named. So what I'm going to do is find the following angle measures. Now I have two angles, ECF and DCE. So I'm just going to do one at a time. So let's go ECF first. So I'm going to go over here. E is over here. C and then F is down here. So ECF. So we're looking at this angle, this blank angle down here. Now what I'm going to do is I'm going to use my other angles there that are written down. I'm going to use those to help me to find ECF. So we have 5x degrees over here. We have 4x plus 22 degrees here. And then down here, we have 70 degrees. Now I want to use my actual angle measures first. So that's 70 degrees. I'm going to look there first to help me out to see if I can figure out what ECF is. You want to use that first because I think that's the easiest. You want to try to stay away from these expressions if you can. If you're forced to use them, you're forced to use them. But in this case, I don't know if we're going to necessarily be forced to use them. OK, so now look here. This angle is 70 degrees. And then this angle down here, ECF, I want to try to figure out what their relationship is. Because if I can figure out what their relationship is, I could see if they're going to be congruent, if they're going to be supplementary, or something to that effect. So notice that 70 is bottom and right, down and right. And then this angle here is also down and to the right. That actually makes those two angles corresponding angles. And if I have a transversal and two parallel lines, corresponding angles are in fact going to be congruent. So this angle, if this angle is 70 degrees, this one down here is also going to be 70 degrees. So that tells me that the measure of ECF is equal to 70 degrees. Now notice there, I didn't have to do any mathematics. I didn't have to do any arithmetic. I just used logic and reasoning. I used deductive reasoning, using my laws and rules and that kind of stuff to figure out what the measure of ECF was. So now let's go to the second one. Let's see if we're lucky enough to be able to use this second one here too. So DCE, DCE, where is that angle at? D here, C and then E. So we're looking for this angle right here, which they tell us is 5x degrees. So I've got to figure out what x is first. I've got to figure that out first. And then after I figure out what x is, plug it back in and I can figure out what that angle measurement is. All right, so now what I have to do is I have to look around to see if this angle has any relationship with anything else. Well, if I look over here, notice that DCE is up into the right. This angle over here is up into the right. So these two are corresponding angles and they are in fact congruent. Just like the last one. Now actually I'm going to set that up as an equation. 5x is equal to 4x plus 22. All right, so if these two angles are congruent, that means I can set them equal to one another and now I'm going to solve. Now actually this is actually pretty easy to solve. So what I'm going to do here is I'm going to take the 4x and subtract it over to the other side, getting 5x minus 4x is 1x. So that means x is equal to 22. So actually it wasn't very difficult at all to find out what the variable was. Now what I have to do is I have to plug it back in. Remember that whenever you solve for something, you're not done, not in geometry. You always got to take it and plug it back in. So if x is equal to 22, that's what I got to plug in right here. So I got to plug in 5 times 22. Let's see, 5 times 20 is 100. 5 times 2 is 10. So this is going to be 110 degrees. There we go, 110 degrees. All righty. After looking at this, after figuring out this example, I see another way to do this. Now notice here, if this is 110 degrees, notice here, this is 110 degrees. Notice down here, these two angles right here. I can't believe I didn't see this initially. These two angles right here are a linear pair. It's another vocab word from previous. They are a linear pair. Linear pairs are supplementary. Supplementary angles add up to 180 degrees. So if this bottom one was 70, this one had to be 110 so that they both added to get 180 degrees. If I would have known that, if I would have seen that right away, I would have said something, but I was still concentrating on using our theorems and postulates. I didn't see that right away. Now what that tells us is that there's going to be many different ways of doing these problems. Not everything is going to be set in stone. Not every way is going to be solved just one way. There's going to be many different ways to do this. So let's do another example. Let's do another example. This one kind of goes the same way around. So I'm going to go a little bit faster through this. A little bit faster through this. Notice here that I have two parallel lines, AC and GD are parallel. In fact, they're supposed to be parallel, but I don't see my parallel symbols here. I must have forgot to write those. Let me just write those in there. This is right here. These are parallel. There we go. They're my parallel symbols. OK, now those are parallel. I forgot to put those in. That's OK. All right, so now what I have to do is I want to solve for EDG and BDG. Now I'm just going to worry one at a time. EDG. So let's look over here. EDG. So it's this bottom one down here. Bottom one down there. So now what I'm going to do is try to figure out how I'm going to solve for that. Looks like I have to solve for X first. I'm going to solve for X first. So I've got to find something else to use. So I can either use this X degrees here, or I can use 2X minus 135. Now notice that bottom left, and then this angle up here, is bottom left. So actually, these two, these big ones here, are in fact congruent. They are in fact congruent. So I can set them equal to one another. So I'm going to do that work down here. I'm going to do that work down here. Actually, let's move some stuff around. Let's move this down. Give myself some space to work. So I know that 2X minus 135 is in fact equal to X minus 30. These are corresponding angles. They're in the same position. Corresponding angles are congruent when I have parallel lines. Parallel line and a transversal, these two are going to be congruent. So what I'm going to do now is I'm going to solve this. Move the X over to the other side. I'm subtracting an X. I get X minus 135 equals negative 30. Don't worry about all the negatives. Everything's going to sort itself out in the end. Now I'm going to add 135 over to the other side. Subtraction here, so I've got to add it over to the other side. So X is equal to 105. Negative 30 and a positive 135, that makes a positive 105. And now that I have what X is equal to, I'm going to plug it in here to figure out what EDG is. EDG. So I'm going to plug that in. So that's going to be 105 minus 30. So that is going to be 75. 75 degrees. So EDG is going to be 75 degrees. I'm going to write that down right here, 75 degrees. Kind of keep track of that. So that's one way to find EDG. Now what I'm going to do is I'm going to find this next one. Oh, it looks like I clicked too much. I'm going to find this next one. BdG is a different color here. BdG. So where is BdG? BdG. OK, so that's this little X right here. Well, did I already find out what X was? Oh, yes, I did. Look over here. The work that I did previously, X is equal to 105. So, oh, awesome. BdG, just double check that. BdG, yep. BdG is just X. I already found out what X is. It's 105. I already did all the work. Yay. I already did all the work. I don't have to do anything else. What's nice about these problems is that they are related to one another. And so the work that you did from a previous problem is still relevant to the other problems that you do, which is really, really nice. Sometimes you do a lot less work for yourself by simply just recognizing that and using work from another problem. Using work from a similar problem here. Now, another way that you could have done this, notice that these two right here are a linear pair. They're a linear pair. So they're supposed to add up to 180 degrees. So these two angles need to add up to 180 degrees. So 75 and 105 do, in fact, add up to 180 degrees. So you could have used that also. Now, what you could have also done is you could have used this X. And you could have used this 2X minus 35. And you could have these are same side interior angles. You could have added them up, set them equal to 180 degrees. It would have looked like this. 2X minus 135 plus X equals 180, first angle, second angle. Or actually, let's do first angle here, second angle here. Here's the second angle. Here's the first angle. You could have done it that way. And you would have solved and you would have gotten the exact same answer. You would have solved and it got the exact same answer. Actually, really quickly, just to prove it, 3X is equal to subtract that over. That would be 545. And then X divided by, whoa, wait a minute. Oh, hold on. Come on, Mr. Man, you're doing this too fast. So here we go. Don't subtract it over. You add it over. So 135 plus 180 is going to be 315. 315, I got to do the right operations. And then divide by 3X is equal to 105. So notice, you would have got the exact same answer. X is equal to 105, which means BDG is 105 degrees. So hopefully, you learned something from that. It doesn't really matter. Sorry, it does matter with the work that you do for your previous problems. Just make sure everything is legible. You write everything out. You understand what's going on. Because your previous problems can really help out with the next problems that you're doing.