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Worked example: Triangle angles (diagram)

We have a right triangle that's split into several smaller right triangles. The angle measures in any triangle always add up to 180 degrees. By using this fact and looking at complementary (adding up to 90 degrees) and supplementary (adding up to 180 degrees) angles, we can find the missing angle measures in the triangles. Created by Sal Khan.

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  • mr pants teal style avatar for user Carson.Alford
    what does the little O with the - beta thing mean??
    (158 votes)
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  • leaf red style avatar for user Jack McClelland
    Everyone is asking about theta, but what if there are two angles you need to figure out? Do you label one theta and the other theta2, or is there a different symbol used?
    (44 votes)
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    • old spice man green style avatar for user newbarker
      You're right. You can use theta1 and theta2 (the 1 and 2 will be a subscript, sometimes spoken "theta sub 1").

      Other perfectly valid ways of representing other angles is to use other Greek letters. Alpha (α) and beta (β) are very often used to represent unknown angles.

      Even the favourite 'x' gets used frequently for angle. It depends on the author and providing there's no ambiguity, whatever letter you choose is fine.
      (57 votes)
  • leaf red style avatar for user Jack McClelland
    Sal's algebra in his first approach for solving the problem reminded me of a question: when doing that sort of algebra, do you treat the numbers individually with the positive/negative sign before it? For example: 90-5=x. Could you treat the 90 as +90 and the 5 as -5?
    (14 votes)
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    • mr pink red style avatar for user Angie
      To answer your question: Yes you could treat the 90 as +90 and the 5 as -5. Try it. -5+90 and you still get the same answer as you would get with 90-5.

      Try another: -80 + 10 which would equal -70. If you flipped the problem but kept the signs it would be (+)10 - 80 which would also give you -70

      hope this helped!!
      (27 votes)
  • aqualine ultimate style avatar for user Vinhee Cho
    Is "beta" the same as "theta?"
    (12 votes)
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  • scuttlebug yellow style avatar for user Perry
    Since when did they use Greek letters? Is the English alphabet not enough for math that is supposedly for numbers?
    (14 votes)
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  • marcimus pink style avatar for user mayamstepanova
    Ima cry why is this kinda hard
    (8 votes)
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  • male robot hal style avatar for user tbarberi4436
    do I learn angles in 5th grade
    (2 votes)
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  • blobby green style avatar for user James Goertz
    what is the formula to find the degrees in a triangle that 180degrees largest is 25 degrees more than the smallest the some of the 2 smallest is 30 degrees more than the largest
    (3 votes)
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    • orange juice squid orange style avatar for user Shawn
      One way to do it is to make a system of equation:
      let a be the largest angle and b, c be the smaller two angles, and a + b + c = 180
      a - 25 = c
      b + c = a + 30
      Notice that b + c = 180 - a
      You can now rewrite the second equation by substituting b + c with 180 - a so that you only have to deal with 1 unknown term:
      180 - a = a + 30
      Rearrange the terms you get:
      2a = 150
      a = 75
      Now using the previous equations you can figure out the rest of the terms:
      c = a - 25 = 75 - 25 = 50
      b = a + 30 - c = 75 + 30 - 50 = 55
      and there you go :D
      a = 75, b = 55, c = 50
      So I guess there is no "formula" to do these questions, but as a matter of fact, there are many ways to approach problems like this. It's all about figuring out a way, like solving a puzzle.
      I hope this helped
      (7 votes)
  • aqualine ultimate style avatar for user Dantas Profile
    Why ''32°'' and ''Y'' have the same degree? I can not find a reason for such equivalence, this happens with the other angles of this problem, please someone help me! ( i find all angles, and they all shows this equivalence ) .
    (3 votes)
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  • piceratops ultimate style avatar for user Challenger68
    Why would you subtract 90 degrees from that equation? I haven't exactly figured that out.
    (3 votes)
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Video transcript

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.