In the figure below, the measure of
angle 𝐵𝐴𝐶 is equal to 30 degrees. If the ray from 𝐴 through 𝐶 is an
angle bisector, what is the measure of angle 𝐵𝐴𝐷?
Let’s begin by identifying first
angle 𝐵𝐴𝐶. It’s the angle enclosed by the line
segments between 𝐵 and 𝐴 and 𝐴 and 𝐶. And so this angle here is 30
degrees. Now, we’re told that the line from
𝐴 and passing through 𝐶 is an angle bisector. Now, a bisector cuts something in
half, so an angle bisector divides an angle exactly in half. And this therefore means that angle
𝐶𝐴𝐷 must be equal to angle 𝐵𝐴𝐶. It’s also 30 degrees.
Now we’re looking to find angle
𝐵𝐴𝐷. That’s this one on our diagram. We can say that it’s equal to the
measure of angle 𝐵𝐴𝐶 plus the measure of angle 𝐶𝐴𝐷. Or alternatively since those angles
are equal, it’s two times the measure of angle 𝐵𝐴𝐶. That’s two times 30, which is equal
to 60 degrees. The measure of angle 𝐵𝐴𝐷 is 60
degrees. In general, we can say that an
angle’s bisector divides it into two equally sized adjacent angles.