 This video is a couple of examples involving trapezoids. Now that we've learned the properties of an isosceles trapezoid we can do this problem. Yoga is isosceles and it has bases Y, O, and G, A. So you'll notice that the picture only has Y labeled. Well you have to remember what bases are in an isosceles trapezoid. So remember the bases are the parallel sides. So because these are the parallel sides I'm going to name because the Y is here I'm going to put the O, G, and the A. And now I'm going to use the information given to me which is that angle Y is 122 degrees and we're going to find the other three angles. Well remember that one of the properties of an isosceles triangle is that the base angles are congruent. Well that means that if angle Y is 122 degrees then angle O has to be 122 degrees. And so we've found the measure of angle O is 122 degrees. To find angles A and G we do know that they're congruent but we have to have a way of figuring out what they are. So what you're going to do is remember back to section 6.1 we know that a quadrilateral, a four sided shape has to add up, the angles have to add up to 360 degrees. So what I'm going to do is I'm going to add 122 plus 122 and that gives me 244 degrees. Then I'm going to subtract that 244 degrees from 360 and we get 116 degrees. So now we know that angles A and G are congruent and so I can take the 116 degrees and divide by 2 and we get 58 degrees. So that means the measure of angle A is equal to the measure of angle G and they are both equal to 58 degrees. In this problem we are told once again that yoga is isosceles. So I'm going to just go ahead and label Y, O, G, A and we are given Y, G and O, A. Well those are the diagonals of this trapezoid and if you remember one of the properties of an isosceles trapezoid is that the diagonals are congruent. So we're going to set X squared minus 8X equal to 9X minus 70 because we know that the diagonals are congruent. So now we have a problem here again that's going to involve factoring to solve because we have both an X squared and an X term. So the first thing I'm going to do is I'm going to subtract the 9X and I'm going to add the 70 to both sides so that we have everything on one side and it's equal to zero. So if I subtract the 9X I'm going to have X squared minus 17X and if I add the 70 it's going to be plus 70 equals zero. So now I'm going to factor that. So what two numbers multiply to 70 that add to negative 17? Hopefully X minus 10 and X minus 7 come to mind. And then I'm going to solve so I get X equals 10 and X equals 7. Now what we have to do is we have to go back to the original information and we have to see if these answers make sense. Now they're both positives so that's good but we want to actually plug them in and see if they make sense in this problem. So if I plug 10 in for X here I get 10 squared minus 8 times 10. Well 10 squared is 100 minus 8 times 10 is 80 so 100 minus 80 is 20. If I plug 10 in over here for X I get 90 minus 70 which is 20. So both of those answers make sense. Then I'm going to try it in using the X equals 7. So if I plug 7 in I get 49 minus 56 and here's where you see the problem. 49 minus 56 is a negative number and of course you can't have a diagonal that's a negative length. So that means X equals 7 does not make sense for this problem and we would only use X equals 10 for our solution.