Exercise 30 Page 39

We have been asked to find the measure of angle $ABC.$ The expression $m\angle ABC$ tells us to find the measure of the angle between rays $BA$ and $BC.$ Similarly, $m\angle ABD$ tells us to find the angle between rays $BA$ and $BD.$ Let's start by identifying the three angles.

It is given that $\overrightarrow{BD}$ is the angle bisector which means that it divides $\angle ABC$ into two congruent angles, $\angle ABD$ and $\angle CBD.$ Therefore the measures of $\angle ABD$ and $\angle CBD$ are equal. Let's substitute the given expressions for those two angles into equation above. Then we can solve for $x.$ We know that the sum of $m\angle ABD$ and $m\angle CBD$ is equal to $m\angle ABC.$ Let's substitute the given expressions for the two smaller angles into equation above. Now, we can substitute $x=12$ in the expression to find $m\angle ABC.$