 We have another theorem when it comes to parallelograms, that if a quadrilateral is a parallelogram, then its diagonals bisect each other. Let's go ahead and draw on the diagonals, which we're going to go from opposite angles, and when the theorem says bisect each other, that means that each diagonal cuts the other one into equal pieces. If we go ahead and we say that this point here is E, where they cross, that means that this segment AE would be congruent to segment EC, and because they bisect each other, DE also is congruent to EB. Keep in mind that all four of these aren't necessarily congruent to each other. This just means that these two will always be congruent and these two will always be congruent. If we go to our example, we can see how we'll use that in solving problems. This problem asks us to solve for X, and it gives us two values of some segments. It's important to go ahead and take these values and label your illustration with them, instead of just assuming that they're going to be equal to each other. Sometimes they might be, but let's go ahead and label our diagram to see how it looks. If I know that AC, this whole piece right here, is X squared, and I know that EC, this segment right here, is X plus 4. That means they're not necessarily equal to each other, and we're going to have to set up an equation based on what's given to us. We're told that this is a parallelogram, so if we're given diagonals in the illustration, it's a good idea to put in those congruency marks, so we know that AE is congruent to EC. And with that being the case, we know if EC is X plus 4, then we can go ahead and also label AE as X plus 4. Now we have enough information to set up an equation, because we have two pieces that are going to equal the whole. The sum of the parts equals the whole, and we can set up our equation using that information. We know now that X plus 4 plus the other X plus 4 is going to equal the whole diagonal of X squared. Now we have an equation, and solving for X, we're going to use our algebra to solve the equation. First, we're going to want to combine like terms on each side. X plus X is 2X, and 4 plus 4 is 8 equal to the X squared. Now if we want to go ahead and solve for X, we should keep in mind that this X squared here is telling us that we'll probably have to factor in order to solve this equation. So whenever you see an exponent, you're going to want to first put your equation in a standard form and know that you'll probably have to factor. And putting it in standard form just means we're going to get everything over to one side and set the equation equal to zero. So when I subtract everything from the one side, I'm going to get X squared minus 2X minus 8 equals zero, and it doesn't matter which way that equation looks. We're still able to factor that now. Now that it's in standard form, you should see that whatever the factor pairs of 8 are that are going to add up to the negative 8 that will add up to the negative 2 will give you the numbers you need to factor that and set that equal to zero. I know this is going to be X and X. If you're unsure of the factor pairs that would work, it would be a good idea to write those out. The factor pairs of 8 are 1 and 8 and 2 and 4. And the combination here that will get us to a negative 2 are going to be the 2 and 4 factor pairs. So I'm going to go ahead and put those values in and then we'll look at the sign. What sign do the 2 and 4 have to be in order to get this to a negative 2? We'll have to have a negative 4 and a plus 2. You could foil this quickly to see that, yes, indeed, this is factored correctly. Mostly that 2X and negative 4X will add up to negative 2 and 2 times negative 4 multiply to negative 8. Our final step then to solve this is to solve for X saying if X plus 2 equals 0 then my X would have to equal negative 2 and if X minus 4 equals 0, my X would have to equal positive 4. So my answer to solve for X, I'm going to have the two answers negative 2 and 4.