 In this video, we're going to be talking about coordinate proofs. Coordinate proofs are different from the flow proofs that we're used to in that we're going to be using some specific values and plugging them into familiar formulas to prove what is asked of us. In this case, we are asked to prove that if we are given a parallelogram, let's show that both pair of opposite sides are parallel. That means we're asked to prove that these two sides are parallel and that these two sides are parallel. When we talk about two segments being parallel, that means these two segments right here would have the same slope and so we are going to be using the slope formula today. It might be a good idea to jot down on the side somewhere this slope formula just as a reminder to you if we haven't used it for a while. Remember that the difference of the y values are on top, difference of the x values on bottom, and this is just remember the symbol for slope is m. We're just going to prove that one slope is equal to another slope. That's what we'll be doing today. You might want to jot down also a couple other formulas that we'll be using in the future in some homework and stuff. In other projects, we may be asked to show that segments bisect each other and in that case to show that diagonals specifically would bisect each other, we would use the midpoint formula and you'll be needing to know that for your homework. So you might want to jot down the midpoint formula that the midpoint formula is the average of the x's and the average of the y's and it's actually going to be a point to show that midpoint of the diagonal. And this will just prove that one midpoint is equal to another midpoint. And then the last one that you might want to jot down for future is the distance formula. The distance formula is going to show that segments are congruent and we know that in parallelograms sometimes we'll need to show that opposite sides are congruent and that's where the distance formula will come in. So go ahead and jot that down so in the future you can show that one distance is equal to another distance. But for today we're going to be using the slope formula and before we do that the problem asks you to assign coordinates. I'm going to give you these coordinates and ask you to place them on your parallelogram. There are four vertices there and in your homework you would have to do this on graph paper and plot each of the points. We don't have graph paper here. When you plot those points you should just go ahead and label each of those and then put those values there as well. It is helpful to put the actual coordinate values up there as we go through and use the slope formula you'll see why that is. Take a minute, pause the video if you need to and label that parallelogram and then we'll talk about what they're asking us for the given and approved statement. The given statement, this part A up here, just says place a parallelogram, L-O-R-I, on the coordinate system. So that means they're giving us the information that we have a parallelogram. In fact they named it parallelogram, LORI. And we're asked to prove that both pairs of opposite sides are parallel. So this is my if then statement and we can go ahead and fill this in. We're given that we have a parallelogram and remember this is the symbol for parallelogram. You can shorten it up. We're given parallelogram, LORI and we're asked to prove both pairs of opposite sides are parallel. So that means I'm trying to show that this segment O-L is parallel to this segment RI and that's what this part of the approved statement is. O-L is parallel. Remember the parallel symbol is just those two vertical lines. O-L is parallel to RI and then at the same time the other opposite sides O-R is parallel to LI. So we need to show and prove that both of those are parallel in order to do this problem. So we said we were going to be using slope formula for this. Specifically, if I want to prove that O-L is parallel to RI I can use the slope formula, plug these values in for these two vertices and show that O-L, this slope, is going to be equal to this slope given this information about the vertices. And then at the same time we'll be finding the slope of this segment and show that it's equal to this segment. So we're actually going to be doing the slope formula four times and you do need to show the work for all four slope values for each of those sides. I'm going to make some more room for myself and it doesn't matter where you start. I'm going to start by finding this slope right here. So I'm going to say the slope, this is the M, the slope of O-L is equal to and then I'm going to plug my values into the slope formula. Remember it's the Y values on top so it's going to be 7 minus 0. That's these two Y values here. We're doing 7 minus 0 on top and then on the bottom because we started with the 7 we have to start on the bottom with the 3. It's important to follow that order, 3 minus 0 on the bottom. So it's the difference of my Y values over the difference of my X values. So the slope of O-L is 7 thirds. Now I'm going to find the slope of that opposite side the slope of R-I and it might be a good idea to pause the video and see if you can do all the remaining three of these slopes and match what is on this video. The Y value is 7 minus 0 on top and then the X values on the bottom 13 minus 10 and we get 7 on top, 3 on the bottom. This should be the case that these two slopes here are going to be the same because that's what we're trying to prove that the slope of O-L is parallel to R-I. So we did show that those two sides have the same slope and therefore they are going to be parallel but we're not done because we have to do the same thing now for the other set of opposite sides and calculate those slopes. So again, it doesn't matter where you start first I'm going to find that slope of O-R so the slope of O-R equals and it doesn't matter what value first I'm going to take point R first 7 minus 7 R the RY value minus the OY value over 13 minus 3 and we get 0 over 10 and remember 0 divided by anything is just going to be 0. That makes sense that this slope is 0 because we know this looks like a horizontal line and we know the slope of any horizontal line will be 0 but we still do have to show even though this looks like this would also be a horizontal line with a slope of 0 we have to go ahead and show the work for the slope of LI as well we're going to do 0 minus 0 the Y value is on top X value is on the bottom and again we get 0 over 10 which is 0 so we showed that these two segments also have the same slope which is what we need to show to prove that O-R is parallel to LI that's all the work that needs to be shown for this problem that you'll see on the bottom here there's this conclusion statement and what you need to do is just conclude what you just did in my conclusion statement I'm going to write both pairs of opposite sides in a parallelogram are parallel where just basically our conclusion statement is just reiterating what we just proved that both pairs of opposite sides in a parallelogram are parallel