Worked example: Triangle angles (diagram)

So in this diagram over here, I have this big triangle. And then I have all these other little triangles inside of this big triangle. And what I want to do is see if I can figure out the measure of this angle right here. And we'll call that measure theta. And they tell us a few other things. You might have seen this symbol before. That means that these are right angles or that they have a measure of 90 degrees. So that's a 90-degree angle, that is a 90-degree angle, and that is a 90-degree angle over there. And they also tell us that this angle over here is 32 degrees. So let's see what we can do. And maybe we can solve this in multiple different ways. That's what's really fun about these is there's multiple ways to solve these problems. So if this angle is theta, we have theta is adjacent to this green angle. And if you add them together, you're going to get this right angle. So this pink angle, theta, plus this green angle must be equal to 90 degrees. When you combine them, you get a right angle. So you could call this one-- its measure is going to be 90 minus beta. And now we have three angles in the triangle, and we just have to solve for theta. Because we know this angle plus this angle plus this angle are going to be equal to 180 degrees. So you have 90 minus theta plus 90 degrees plus 32 degrees-- so I'm going to do that in a different color-- is going to be equal to 180 degrees. The sum of the measures of the angle inside of a triangle add up to 180 degrees. That's all we're doing over here. And so let's see if we can simplify this a little bit. So these two guys-- 90 plus 90's going to be 180, so you get 180 minus theta plus 32 is equal to 180 degrees. And then what else do we have? We have 180 on both sides. We can subtract that from both sides. So that cancels out. That goes to 0. And then you have negative theta plus 32 degrees is equal to 0. You can add theta to both sides. And you get 32 degrees is equal to theta, or theta is equal to 32 degrees. So it's going to actually be the same measure as this angle right over here. That's one way to do the problem. There's other ways that we could have done the problem. Actually, there's a ton of ways we could have done this. We could have looked at this big triangle over here. And we could've said, look. If this is 90 degrees over here, this is 32 degrees over here, this angle up here is going to be 180 minus 90 degrees minus 32 degrees. Because they all have to add up to 180 degrees. And I just kind of skipped a step there. Actually, let me not skip a step. Let me call this x. If we call the measure of that angle x, we would have x plus 90. I'm looking at the biggest triangle in this diagram right here. x plus 90 plus 32 is going to be equal to 180 degrees. And so if you subtract 90 and 32 from both sides. So if you subtract 90 from both sides, you get x plus 32 is equal to 90. And then if you subtract 32 from both sides, you get x is equal to-- what is this-- 58 degrees. Fair enough. Now, what else can we figure out? Well, if this angle over here is a right angle-- and I'm just redoing the problem over again just to show you that there's multiple ways to get the answer. We were given that this is a right angle. If that is 90 degrees, then this angle over here is supplementary to it, and it also has to be 90 degrees. So then we have this angle plus 90 degrees plus this angle have to equal 180. Maybe we could call that y. So y plus 58 plus 90 is equal to 180. You can subtract 90 from both sides. Subtract 90 from both sides. This will become 90. Subtract 58 from both sides, you get y is equal to 32 degrees. Well, if y is 32 degrees, it is complementary. It is complementary to this angle right over here. It is complementary-- let me do that in a new color, not supplementary. It is complementary. It adds up to 90 degrees. It is complementary to this angle over here. We could call it z. So these two combined are going to add up to 90 degrees, or z is going to be equal to 58 degrees. And now we're inside the triangle that we care about to figure out theta, theta that we've already figured out earlier in this video. Well, this z is 58 degrees. If this angle over here is 90, then this one over here is also going to be 90, because they're supplementary. So you have 58 degrees. I wanted to do that in that orange color. So if you have 58 degrees, so you have 58 plus this 90, plus 90, plus theta now is going to be equal to 180 degrees. You can subtract 90 from both sides. That becomes 90, and then you have 58 plus theta is equal to 90. Subtract 58 from both sides. You get theta is equal to 32 degrees again. And so we've got the same answer. I just wanted to do that, show you that there are multiple ways to do these problems. And as long as you're doing things that are logically consistent, you're making assumptions that you can make and then logically deducing step by step, there's multiple ways to get that right answer.