Video Transcript
Olivia is experimenting with the
exterior angles of a triangle. She colors the angles, cuts them
out, and sticks them together as seen in the figure. Will angle 𝑧 fit in the space to
complete the circle?
One way we can answer this question
is to simply imagine the triangle getting smaller and smaller and smaller. We notice that as the triangle
shrinks to nothing, these lines appear to circle around a point. We know that angles around a point
sum to 360 degrees, so we can infer that angles 𝑥, 𝑦, and 𝑧 fit perfectly to
complete the circle. And so, we could add in angles 𝑧
on our original diagram as shown.
But is there a better way to prove
this? Well, yes, there is. We know the angles on a straight
line are supplementary. They add to 180 degrees. We also know the same about
interior angles in a triangle; they add up to 180 degrees. So, we’re going to fill in the
missing three angles in our first triangle. This first one must be 180 minus 𝑥
degrees. This must be 180 minus 𝑦. And over here, we have 180 minus 𝑧
degrees. We know that the sum of these three
interior angles is 180 degrees, so we can say 180 minus 𝑥 plus 180 minus 𝑦 plus
180 minus 𝑧 equals 180. This simplifies to 540 minus 𝑥
minus 𝑦 minus 𝑧 equals 180.
We’re going to subtract 180 from
both sides so that 360 minus 𝑥 minus 𝑦 minus 𝑧 equals zero. And then we’re going to add 𝑥, 𝑦,
and 𝑧 to both sides so that 360 equals 𝑥 plus 𝑦 plus 𝑧. We’ve actually proven that the
exterior angles 𝑥, 𝑦, and 𝑧 do indeed add to 360 degrees. And so, angle 𝑧 will fit in the
space and form a perfect circle. And so, the answer is yes. Note that we can generalize this
and say that the exterior angles of any polygon sum to 360 degrees no matter the
number of sides it has.
