Find the measure of angle
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.