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## Integrated math 1

### Unit 17: Lesson 4

Theorems concerning triangle properties# Angles in a triangle sum to 180° proof

CCSS.Math: ,

Learn the formal proof that shows the measures of interior angles of a triangle sum to 180°. Created by Sal Khan.

## Want to join the conversation?

- Im in 7th grade. When i started it was hard I think the way I learned from my teacher is harder because I cant ask the teacher to repeat it or pause soi can write the problem down but when he assigned me this while the highschoolers had a field trip. khan academy's is WAYYYYYYYYYYYYYYYYYYy....
**WAY*100**easier and more fun.(5 votes) - At0:01, Sal mentions that he has "drawn an arbitrary triangle." What is an arbitrary triangle?(3 votes)
- Arbitary just means random. Sal means he just drew a random triangle with sides of random length.(2 votes)

- At0:25, Sal states that we are using our knowledge of transversals of parallel lines. What does that mean?(4 votes)
- A transversal is a line that intersects a pair of parallel lines. The angles that are formed between the transversal and parallel lines have a defined relationship, and that is what Sal uses a lot in this proof. You can learn about the relationships here: https://www.khanacademy.org/math/basic-geo/basic-geo-angle/angles-between-lines/v/angles-formed-by-parallel-lines-and-transversals(3 votes)

- why cant i fly(3 votes)
- Isaac Newton(2 votes)

- You proved "corresponding angles are equal" using "angles in a triangle sum to 180"[https://www.khanacademy.org/math/geometry-home/geometry-angles/geometry-angles-between-lines/v/proof-corresponding-angle-equivalence-implies-parallel-lines]. Now here you are proving "angles in a triangle sum to 180" using "corresponding angles are equal". Isn't this circular logic?(2 votes)
- Yes, but keep in mind that we already know that both of these are true. We're just showing how we know.(3 votes)

- What is an abbertary triangle? I know I spelled it wrong(2 votes)
- arbitrary

ˈ/ɑːbɪt(rə)ri/

adjective

1.

based on random choice or personal whim, rather than any reason or system.

In this video it means any triangle you care to think of.(2 votes)

- what is a median and altitude in a triangle(2 votes)
- A median in a triangle is a line segment that connects any vertex of the triangle to the midpoint of the opposite side. An altitude in a triangle is a line segment starting at any vertex and is perpendicular to the opposite side.(2 votes)

- I get confused when dealing with supplementary and just the equaling thing anyone got a tip? to help unconfuse me(2 votes)
- Sure!

Supplementary angles are adjacent angles that, when added, equal 180 degrees. Since the sum of supplementary angles equal 180 degrees, they create a straight line. Complementary angles, however, are adjacent angles that have a sum of 90 degrees. These types of angles make a "corner" shape.(2 votes)

- At0:03, Sal mentioned 'interior angles'. What are interior angles?(2 votes)
- If you have two lines cut by a transversal, interior angles are angles inside those two lines.(2 votes)

- This is interesting. However, isn't the point of the video to figure out what x,y, and z are worth?(1 vote)
- No, the point is to show that x+y+z=180º. This is an arbitrary triangle, which means that x, y, and z can be any value between 0 and 180.(4 votes)

## Video transcript

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.