Worked example: Triangle angles (diagram)

so 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 the use 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 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 angle is adjacent to this green angle and if you add them together you're going to get this right angle you're going to get this right angle so this this this pink angle theta plus this green angle must be equal to 90 degrees they when you combine them you get a right angle so you could call this one this measure is going to be 90 minus theta and now we have 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 so plus 90 degrees plus 90 degrees plus 32 degrees so let me do that in a different color plus 32 degrees plus 32 degrees 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 if we subtract so these two guys 90 plus 90 is going to be 180 so you get 180 minus theta plus 32 is equal to 180 degrees and then we have 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 zero and you have negative theta plus 32 degrees is equal to zero you can add theta to both sides and you get 32 degrees is equal to theta or theta is equal to thirty-two degrees so it's going to actually be the same it's going to 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 we could have said and actually there's a ton of ways we can done this we could have looked at this big triangle over here we can say 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 obviously this let me actually not skip a step let me call this X if we call that the measure of that angle X we would have X plus 90 I'm looking at this the biggest triangle in this diagram right here X plus 90 plus 32 plus 32 is going to be equal to 180 degrees and so you 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 X is equal to 58 degrees X is equal to 58 degrees fair enough now what else can we figure out well if this angle over here 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 it also has to be 90 degrees so then we have this angle plus 90 degrees Plus this angle 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 Y is equal to 32 degrees well if Y is 32 degrees it is it is complementary it adds up it is complementary to this angle right over here it is complementary let me do that in a new color not supplementary dis complimentary 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 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 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 get 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 you're making assumptions that you can make and then logically deducing step by step there's multiple ways to get that right answer