 Label the missing coordinates of the following parallelograms. We're given four parallelograms here. These are going to be four separate problems. And we're going to use the information given to us about some of the vertices to find the missing coordinates. And this will lead us into doing more with coordinate proof. Keep in mind that this is all about horizontal measures and vertical measures. And you might want to make a note to yourself that when I talk about horizontal measures, that's going to be all of the values that the x-coordinate of your vertices of your vertices represents. So I'm just going to make a note to myself that the x-value is the horizontal measure. I'm even going to put just a horizontal line there and then the y-values of each of my vertices is going to represent the vertical measure. And that is the up and down vertical measure because this is the y-axis. We also want to keep in mind that when we're doing coordinate proofs with parallelograms, if we remember the properties that we've learned about parallelograms, we know that opposite sides are congruent and parallel. That's going to help us do these problems. I have this diagram here that will help us with this. It turns into a symmetrical illustration if we know that opposite sides of parallelograms are congruent and parallel and this measurement is going to be congruent to this measurement. Keep that in mind as we start this problem. So I'm going to go ahead and start with the horizontal piece of the parallelogram, specifically this value of 5. I know that that represents this distance from the origin to this point is 5. I could also go ahead and do that same distance up here because I know that that will be 5 as well. It might be helpful to go ahead and draw these imaginary lines in here and we'll see why that is once we do the other horizontal values. I also know this 3 value represents the distance from the origin to this point. This is my vertices, so that's my horizontal value is 3 and based on what we were just talking about the symmetry, we know that this value up here is 3 as well. Now that we have all of the horizontal pieces labeled, we can see that this point right here represents this whole distance, which is 5 plus 3. That visual can help you see that this horizontal distance is the 5 from this 5 below plus that additional 3 that is congruent to this piece down there. So we know that this x value right here represents the horizontal distance of 5 plus 3 equals 8. In these parallelograms, it is always going to be true because of the properties of being congruent in parallel that the x value is always going to be this value plus this value to get this missing piece right here, 3 plus 5 equals 8. Let's move on to the vertical distance now and the vertical distance is a little more straightforward. We're going to use the fact that this y value is 4 that represents this vertical distance right here and of course if that's 4 then every vertical piece along the way is also 4. So we know that the y value of our missing coordinate is also going to be 4. A good check at the end is to just start at the origin and go to my missing point. If I'm starting at the origin, I'm going to go out 5 plus 3 is 8, go out horizontal and then go up 4 and that will confirm that this indeed is my answer. Another thing to keep in mind is parallelograms this distance is always going to be the same so the y values will always match up on top and the y values on the bottom will always match up because they are parallel and that vertical distance will be the same above and below. Let's go ahead and find missing coordinate problem. I want you to pause the video right now and see if you can do this one on your own. Okay, if you try this we'll go through the same thing that we just did in that we'll start with the horizontal pieces. I know that this distance from here to here is C units long. We're just using variables now. It's the same concept we were just doing. We know that this distance, if you want to fill in this distance, is also C units long. I'm going to go ahead and fill in these imaginary vertical lines to help me see the symmetry there and then we'll talk about the a value. I know that this distance right here is a units long, which means this distance right here is also going to be a units long. That means that this entire distance, this entire horizontal distance, is going to be C plus a and that's the answer for the x value of that. We can't simplify that. It's just going to be those two variables added together. When we talk about our vertical distance we know that this vertical piece is represented by b and so the vertical piece right here is also going to be b. Remember these values, the y values on top will match and the y values below will match. This is a good check before you move on and then we know that the x value of this vertices up here is always going to be this x value plus this x value, a plus c. You could write this either as C plus a or a plus c. It makes no difference. Let's move on to the third one and this one is set up where now we have to find this missing value. Same concepts though, so let's go ahead and fill in what we know. I'm going to just put in these imaginary lines for myself to help see the symmetry there and then again, I'm just going to start with this value right here. That means this distance from the origin to this vertices below is 8 units long and I'm going to go ahead and fill that out up here just to help me visually see that. Now this time I don't know what this little piece right here is measured. I'm given the total over here, so I'm going to work a little bit backwards. I know this total distance from here to here, to this vertices right here, is 11 and I know I get that 11 by saying 8 plus this piece equals 11. So I'm trying to find out that missing piece. 8 plus what equals 11? If I go ahead and write that, you could do that in your head. I know we know that x is going to equal 3. I'm just finding this little piece right here is 3. That means this little piece down here is also 3. I just work backwards from the first two problems. So I know the x value of this missing coordinate is 3 and I can verify that by remembering that this plus this 3 plus 8 will equal that upper right-hand vertices 3 plus 8 equals 11. Now let's move on to the vertical piece. The vertical piece in parallelograms is always a little bit easier. We know that this vertical distance is 6 and so this vertical distance is also 6 which gives us that missing piece right there. We can also, a good idea to just check at the end again. We know these y values should be the same. 6 and 6, these y values should be the same, 0 and 0 and then again 3 plus 8, this plus this equals that upper right-hand value. Go ahead the last problem. Try one more time to hit pause and see if you can fill that out. This is the trickiest of them, but let's see what we can do. Hit the pause button and see if you can fill these in. I'm going to approach this. Like I do of filling in what I know first. I know that x value is r. That means this distance right here is r and I'm going to fill in my little missing lines there. This value right here is also r. What else do I know about this? That like the last problem, they give us this whole distance from here to here, to this vertices right here, is n. And so just like the last problem, we need to find the difference of n and r. Now we just have variables instead of real numbers. So we know that this chunk right here is what we're trying to find this little horizontal distance is going to be the whole horizontal distance n minus r and that will give us our missing piece there. That's as simplified as it can get because they're not real numbers. We're going to leave those variables just as n minus r. And that makes sense if we talk about our check that we do at the end. We know that this value plus this value has to equal that top right coordinate. We could even write that out to make sure that that comes out correctly in that n minus r plus r equals n. All I'm doing is seeing if this plus this does equal that, that top right coordinate. And if I do that n minus r plus r, these two cancel and I get n equals n. So I know that I have the correct answer there. And let's finish this off by finding the vertical distance now. We know this vertical distance is m. And so this vertical distance of course is also m and we can fill in that missing y value of m. Again, I can just do a double check at the end. These y values are the same. These y values, oh, we didn't fill in that y value right here. If this y value is zero, then this y value also is zero. And we can go ahead and fill that in. And then again, let's just talk about our x values n minus r plus r equals n and we show that that is correct over here and filled in our missing coordinates.