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

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.

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