 With this problem, it's tough to know where to start. I've got a whole lot of variables. I've got y. I've got x. I've got n. I've got r. I've got t. So many variables. It's hard to know where to start. What I would recommend is take a look at this triangle. Triangle ABC. In that triangle, we have two angles where y is the variable. And so that kind of gives you a little clue that y is sort of the starting point for this triangle, this similarity problem. So let's take a look at these angles, A, B, and C. The triangle sum theorem says that all three of those angles have to add up to 180. So we know angle A, B, and C add up to 180. So therefore we get this new equation. So let's combine like terms. If we subtract 45 from both sides and then divide both sides by 9, we get that y is equal to 15. And so that piece of information is kind of a linchpin, sort of a key to finding out a lot of other side lengths. Sorry, not side lengths, angles. I know the measure of angle A is 74 because y is 15, you can substitute that value in for 5y minus 1. We can do the same thing for angle B. We know B is equal to 68 degrees because if we substitute in 15, we get 68. And now we can use that information to help us find some other angles. For example, let's take a look at angle D. Angle D must be equal to angle A. And the reason why is since these two triangles are similar, A corresponds with D. And so that makes their angles equal. And so 3t plus 11 must equal 74. When you solve that, you get that t is equal to 21. Now we can do the same thing for R. I see R is related to angle F. Now F corresponds with angle C. Since those angles correspond, they must be equal. And so now we can say 2R plus 4 equals 38. When you solve that, you get that R is equal to 17. So now the only thing that's left are these side lengths. In particular, we're missing angle, not angle, sorry. Side length x, 3x plus 2, and n in 2n plus 2. So I'm going to clear the page and let's take a look at side lengths. So let's ignore the angle information for now. I need to find what x and what n equal. In order to do that, I have to use a known ratio. So let's first solve for x. The side 3x plus 2 refers to AC. So let's use AC as a ratio. I know that AC is the first and third letter of triangle ABC. So therefore AC corresponds with DF. That's our unknown ratio. And we want to set that ratio equal to unknown ratio, so we can use proportions. The only known ratio here is AB, which is first and second, and DE. So in order to set up my proper ratio, I'll have to set up AB on top divided by DE down below. So now we need to cross-multiply. 12 times the quantity 3x plus 2 is equal to 15 times 16, which is 240. The first thing to do to solve that will be to distribute 12, then subtract 24 from both sides, and then divide by 36 on both sides, and we get that x is equal to 6. So now we found x. The last thing that we need to find is n. Since the only thing that we need to find is n, I'm going to cover up a bunch of this stuff. We already know what y is. We already know what x is. We already know r and t. And so the only thing left is to find n. And n refers to the length fe. The corresponding side length of fe, if you look at the last two letters of triangle def, corresponds with the last two letters of triangle abc. So there's our unknown ratio. In order to solve that, we'll need to set that ratio equal to a known ratio. The only known ratio we have, remember, was 15 and 12. So DE and AB. Now since I have fe on the top of this ratio, and fe is the second triangle, I need to put the second triangle on top in the next ratio. So DE corresponds with AB. Cross multiply. Let's distribute the 15, and now we're done.