Main content

### Course: 8th grade > Unit 5

Lesson 2: Triangle angles- Angles in a triangle sum to 180° proof
- Find angles in triangles
- Isosceles & equilateral triangles problems
- Find angles in isosceles triangles
- Triangle exterior angle example
- Worked example: Triangle angles (intersecting lines)
- Worked example: Triangle angles (diagram)
- Finding angle measures using triangles
- Triangle angle challenge problem
- Triangle angle challenge problem 2
- Triangle angles review

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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.

## Want to join the conversation?

- what does the little O with the - beta thing mean??(156 votes)
- theta is the greek letter representing the unknown. In math, it represents an unknown angle whereas a letter represents a variable.

In trigonometry, theta is used frequently(285 votes)

- 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?(42 votes)
- 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.(56 votes)

- 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)- 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!!(25 votes)

- Is "beta" the same as "theta?"(12 votes)
- Beta and Theta are 2 different Greek letters, but they can both be used as variables, like x & y for angles. If you are asking whether it matter which one you use as a variable, it doesn't matter, they both work. But, if you are asking whether they are the same letter, they are not the same.(7 votes)

- I have a grapefruit rn

who wants it?(13 votes)- the comment section did not dissapoint XD(2 votes)

- where did the o come form(8 votes)
- That's not an "o". That's the greek symbol θ, or theta. I don't know why Sal chose this specific letter for this example, but Sal does as Sal wants.(9 votes)

- Since when did they use Greek letters? Is the English alphabet not enough for math that is supposedly for numbers?(11 votes)
- Ima cry why is this kinda hard(6 votes)
- do I learn angles in 5th grade(2 votes)
- interesting... But will actually have problems were the triangle is split into different parts or it is always like that?(2 votes)

- 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)
- 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)

## 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.