Question Video: Finding the Size of an Exterior Angle of a Triangle Given Its Supplementary Angle’s Size Mathematics

Video Transcript

Find the measure of angle 𝑋𝐴𝐶.

Angle 𝑋𝐴𝐶 is this exterior angle. It’s an angle that’s on the outside of a polygon. Now, it has to be one of the polygon’s sides extended and then the angle adjacent to it. And it looks like we have two other exterior angles and we do. The bottom side is extended and we have two exterior angles.

All exterior angles of a polygon should add to 360 degrees. So the measure of angle 𝑋𝐴𝐶 plus the measure of angle 𝑍𝐶𝐴, the measure of angle 𝐴𝐵𝑌 should equal 360 degrees. Now, we know two of these angle measures. So plugging them in, we can solve. So 121 plus 131 is equal to 152 degrees. So to solve for our angle the measure of angle 𝑋𝐴𝐶, we need to subtract 152 degrees from both sides of the equation, resulting in 108 degrees.

Now, there are other ways to solve this. Let’s just try one of the ways. We could use the inside angles of the triangle. And a triangle adds to 360 degrees, the interior angles of it. So these three angles add to 180 degrees. But how do we get them?

Well, a straight line makes 180 degrees. So the angles next to each other that make a straight line are called supplementary angles. They’re adjacent, meaning next to each other. And they make a full straight line. So if they should add to 180 degrees, we can subtract 121 from 180. And we can find the measure of angle 𝐴𝐶𝐵. And after substracting, we find that it is equal to 59 degrees.

And now, we can repeat the process to find the measure of angle 𝐴𝐵𝐶 because these angles are also supplementary. So we take 180 and subtract 131. And we find that this angle — the measure of angle 𝐴𝐵𝐶 — is equal to 49 degrees.

So now, we have two of the three interior angles. So we need to add them together and then subtract it from 180 because all of the angles on the inside of the triangle add to 180 degrees.

So the measure of angle 𝐴𝐶𝐵 is equal to 59 degrees. The measure of angle 𝐴𝐵𝐶 is equal to 49 degrees. And now, we add them together and their sum is 108 degrees. Now, we subtract it from 180, both sides of the equation. And we find the measure of angle 𝐶𝐴𝐵 is equal to 72 degrees.

Now, this is useful because that angle and the angle that we want — the measure of angle 𝑋𝐴𝐶 — they should add to 180 because they’re supplementary. So 180 degrees minus 72 degrees gives us 108 degrees, just like we found before. So that’s a second way.

Now, notice the 108 degree shows up here in the equation is the sum of the measures of angles 𝐴𝐶𝐵 and 𝐴𝐵𝐶. And the reason why is because the two interior angles that are not next to the exterior angle should add to be the exterior angles. So 59 plus 49 is equal to 108.

So whichever way we choose, we find that the measure of angle 𝑋𝐴𝐶 is equal to 108 degrees.