Olivia is experimenting with the exterior angles of a triangle. She colours 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?
Well, we actually know that the sum of the exterior angles of a triangle is equal to 360 degrees. So this actually suggests that yes, angle 𝑧 will fit in the space, because we’ve actually got the angles around a point which also equals 360 degrees. But let’s see if we can actually prove this and work this out.
So what I’ve done is I’ve actually marked the interior angles of our triangle on. So we’ve got 𝑎, 𝑏, and 𝑐. Well, we know that 𝑎 plus 𝑏 plus 𝑐 must be equal to 180 degrees because the interior angles of a triangle add up to 180. And we also know that the angles on a straight line is equal to 180 degrees. So therefore, we can actually write out what 𝑎, 𝑏, and 𝑐 are in terms of 𝑥, 𝑦, and 𝑧. Well, 𝑎 is 180 minus 𝑥. 𝑏 is 180 minus 𝑦. And 𝑐 is 180 minus 𝑧.
Okay, so now let’s try to add these together. Well, therefore, we can say that as 𝑎 plus 𝑏 plus 𝑐 is equal to 180 degrees, then therefore 180 minus 𝑥 plus 180 minus 𝑦 plus 180 minus 𝑧 must also be equal to 180. So therefore, if we can actually simplify, we get 540 minus 𝑥 minus 𝑦 minus 𝑧 equals 180. So then if we actually add 𝑥, 𝑦, and 𝑧 to both sides of the equation, we get 540 is equal to 𝑥 plus 𝑦 plus 𝑧 plus 180.
And therefore, if we subtract 180 from each side, we get 360 is equal to 𝑥 plus 𝑦 plus 𝑧. So therefore, yes this proves what we said at the beginning. The sum of the exterior angles of a triangle is equal to 360 degrees. And then as we said earlier, the angles around the point equal 364. So therefore, angle 𝑧 will fit in the space.