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Problem 29 Hard Difficulty

ACT/SAT In rectangle $A B C D$ , what is $x+y$ in terms of $z ?$
\begin{array}{l}{\text { A } 90+z} \\ {\text { B } 190-z} \\ {\text { C } 180+z} \\ {\text { D } 270-z}\end{array}




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Video Transcript

All right, So we're being asked to figure out what X plus why would be equal to in terms of the on this? Um right now, I don't really have any way to relate. XTO Why in this figure? So instead, what I'm going to do is start working through things that I do know and then see if I can figure out a way to relate them to each other later on. OK, so while I don't know what x and y are with each other, I do know we're told this is a rectangle, and we know that all corners of a rectangle are going to be 90 degrees. Okay, so we at least know that much, That this is 90 degrees and this is 90 degrees. Okay, So what I see when I look at this, I see a triangle right here. My lines are going to be perfect, but, you know, see a triangle right here, and I see triangle right here. Okay. No, we know all triangles add up to 108 degrees, right? All three angles in a triangle add up to 180 degrees. So if we look at Let's look at the Blue Triangle, Ok? If we look at the Blue Triangle here, I know that I have a 90 degree angle down here at the bottom. If the angles in the triangle add up to 180 degrees and I have 90 degrees right here, that only leaves me 90 more to have to deal with. So do you angle? Why right over here and this angle up here that we haven't labeled, So let's just label it h just to give it something. I know that angle. Why an angle age? When we add those together, those must equal 90 degrees. Okay, just by rule of all angles and triangle have to add a 280 degrees. So then, if we look at this green triangle again, we have a 90 degree angle right here. Okay, If that angle is 90 degrees, that only leaves his 90 degrees left to play with. So again, I've got angle X down here and this other angle that isn't labeled, so we'll just call it okay, Angle X and angle K. If we add those together, those also must be 90 degrees, right? Because then 90 plus 90 degrees from X and K, plus the 90 degrees from angle. They would give me the 180 degrees of that triangle. All right, Last equation for us to right here. If every corner of our rectangle is supposed to be 90 degrees, that means this corner up here angle B is also 90 degrees. Meaning, I could say that K plus Z plus age would all equal 90 degrees together. All right, all three of these equations that we've written here are equal to 90 degrees, and the variables air all represent the same thing in each of them. So what we have done is we have created ourselves a system of equations, okay? Meaning I could stack all three of these on top of each other. Now, I know that eventually I'm supposed to be figuring out what x plus y equals. So I'm gonna need tohave x plus y together. So what I'm gonna do is before I stack all three of them together, I'm going to go ahead and add these two equations together so that I can get X plus y right, cause I can see right here if I had these two together immediately. I'm taking why plus X when she all right is X plus Y. I also have a church and K that are being added as well. And then 90 plus 90 would equal 180 degrees. Okay, so now I'm gonna drag this equation down here, stack on top of each other. I'm gonna line up the variables Z has no variable tow line up with, but I do have a plus H and I do have a plus K, and that equals 90 degrees. All right, so here's my systems of equations that I'm working with now. I want to look at everything below this purple line. Everything below this purple. OK, if you've solved systems of equations before then the idea is we want to, uh, combine these together. But we want to try and cancel out some variables. Right? Specifically, I see that I have hnk in both the equations. So why don't we try and get rid of those meaning I want to subtract thes two equations. Okay. So X would be minus nothing because there are no exits down here. Why would also be minus nothing? Because there's no wise down there. Ze would be a negative Z because we are subtracting it right here we are subtracting the Z, but there's no other Z's to combine with. However H minus age would cancel K minus. K would cancel. And then finally, 180 minus 90 would give me 90 degrees Okay for him to put the degrees there. So now I've got X plus. Why minus Z equals 90? Well, let's remember what the question was asking us for. It wanted to know what X plus y equals meaning. I want to get X plus y by itself on one side of the equation. So to do that, I would simply need to add Z over to the other side. And that would give me that X plus y is equal than 90 plus z, meaning If we're being asked what does X plus y equal well, it equals 90 plus Z which, according to our question, would be answered a

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