# Proof: Sum of measures of angles in a triangle are 180 | Geometry | Khan Academy

I’ve drawn an arbitrary

triangle right over here. And I’ve labeled the measures

of the interior angles. The measure of this angle is x. This one’s y. This one is z. And what I want to

prove is that the sum of the measures of the interior

angles of a triangle, that x plus y plus z is

equal to 180 degrees. And the way that

I’m going to do it is using our knowledge

of parallel lines, or transversals

of parallel lines, and corresponding angles. And to do that,

I’m going to extend each of these sides of the

triangle, which right now are line segments, but

extend them into lines. So this side down

here, if I keep going on and on forever

in the same directions, then now all of a sudden

I have an orange line. And what I want to do is

construct another line that is parallel to

the orange line that goes through this vertex of

the triangle right over here. And I can always do that. I could just start

from this point, and go in the same

direction as this line, and I will never intersect. I’m not getting any closer or

further away from that line. So I’m never going to

intersect that line. So these two lines right

over here are parallel. This is parallel to that. Now I’m going to

go to the other two sides of my original triangle

and extend them into lines. So I’m going to extend

this one into a line. So, do that as neatly as I can. So I’m going to extend

that into a line. And you see that this is clearly

a transversal of these two parallel lines. Now if we have a transversal

here of two parallel lines, then we must have some

corresponding angles. And we see that

this angle is formed when the transversal intersects

the bottom orange line. Well what’s the

corresponding angle when the transversal

intersects this top blue line? What’s the angle on the top

right of the intersection? Angle on the top right of the

intersection must also be x. The other thing that

pops out at you, is there’s another

vertical angle with x, another angle that

must be equivalent. On the opposite side

of this intersection, you have this angle

right over here. These two angles are vertical. So if this has measure

x, then this one must have measure x as well. Let’s do the same thing with

the last side of the triangle that we have not

extended into a line yet. So let’s do that. So if we take this one. So we just keep going. So it becomes a line. So now it becomes a transversal

of the two parallel lines just like the magenta line did. And we say, hey look this

angle y right over here, this angle is formed from the

intersection of the transversal on the bottom parallel line. What angle to

correspond to up here? Well this is kind of on the

left side of the intersection. It corresponds to this

angle right over here, where the green line,

the green transversal intersects the

blue parallel line. Well what angle

is vertical to it? Well, this angle. So this is going to

have measure y as well. So now we’re really at the

home stretch of our proof because we will see that

the measure– we have this angle and this angle. This has measure angle x. This has measure z. They’re both adjacent angles. If we take the two outer

rays that form the angle, and we think about this

angle right over here, what’s this measure of this

wide angle right over there? Well, it’s going to be x plus z. And that angle is supplementary

to this angle right over here that has measure y. So the measure of

x– the measure of this wide angle,

which is x plus z, plus the measure of this

magenta angle, which is y, must be equal to 180

degrees because these two angles are supplementary. So x– so the measure of

the wide angle, x plus z, plus the measure of the

magenta angle, which is supplementary

to the wide angle, it must be equal to 180 degrees

because they are supplementary. Well we could just

reorder this if we want to put in

alphabetical order. But we’ve just

completed our proof. The measure of the

interior angles of the triangle,

x plus z plus y. We could write this

as x plus y plus z if the lack of

alphabetical order is making you uncomfortable. We could just rewrite

this as x plus y plus z is equal to 180 degrees. And we are done.

secont

Mindblown.

how did i subscribe to this?

The proof ─ The Alternate Interior Angles of Parallel Lines are equal. in Chinese↓

watch?v=IZmdhZeTWlI

I needed to see this proof. It's funny how things click!

Didn't he already make this video?

porn

b

Nice. Now do it on a sphere… 😉

easy as 1. 2. 3…

Are there different natures to a supplementary angle? It is really just a label for a set of angles that add up to a line, correct?

fuck…

Do you add this new videos you are making to playlist on the khanacademy site?

why is that cool news??.You've got the wrong video to post that on.!

My curiosity Solved, thaxxxx

khan you are the best of the best

we can also prove it by using a circle that contains the triangle .

Missing was to explain the term 'supplementary' and why it has to =180 …since it's been years since I've thought about geometry!

How?

At first I read the tittle: "Proof that the sun is a triangle"

Even though my native language isn't English , I understand clearly what you are said . That's because it's so easy , so thank u so much

I mean you make the math so easy

…. Using alt angles are definitely easier

if you get the chance you should check out desmos online calculator

I added the angles around the top point in the triangle. These angle collectively form a circle (with has 360 degrees) so 2x+2y+2z=360. Dived both sides of the equation by 2 and you get x+y+z=180

what if the angle is blue?

😛

LOL

awesome

ok….

Oh I get it

Can you proof Fermat's last theorem ?

If the angle is Blue, we all know that Red + Blue = Green, so we subtract Blue from both sides and we get Red = Green – Blue. Therefore, if the angle is Blue, the answer is Red. It's just simple algebra! lol!

…

This shit is very boring you need to stop this crap and dont post this bull shit its very very boring 😑😑😑😑😑😑😑😑😑😑😑😑

Where can I find the proof of the equivalence of angles?

Excelent video, very simply explained.

I'm not getting any audio from this video, anyone know why?

X + Y + Zee

Go learn the alphabet

For some reason I was troubled thinking about this last night. Thank you so much!

I'm 17 btw Ap calculus BC. 🙂

for all you people out there who take rules like this to be set in stone i hate to tell you but they aren't. the angles can add up to more than 180 it was proved hundreds of years ago.

Oog… at around 0:45 I got a little dizzy… ugh.

I'd say that this proof only shows the relation of the sum of angles shared by any arbitrary triangle and a half circle. If the half circle was determined as 200 degrees, then it follows – in the same way that you have shown here – that the sum of the angles in a triangle also must be 200 degrees. Right?

My mind was blown at the end of the video thanks! I`m 25 years old and I`ve always “hated“ math when I was younger but now I`m learning to love it! Thank you very much!! You are a gift to humanity.

Thank you. Tomorrow is my exam and now I shall be able to do it

I'm trying to learn how to construct proofs so, thank you for opening my mind!

This was amazing, thank you!

Wow, this video is amazing.

I hate maths specially geometry but this video is very helpful. I Love it

thanks it worked.

boring