Question 308·Hard·Math

Let $a$ be the measure of $\angle BAC$. Because $BC$ extends through $C$ to $E$, $\angle ACE$ is an exterior angle. An exterior angle equals the two far interior angles added together. Since $D$ is on $BC$, $\angle ABC$ is the given $31^\circ$ angle at $B$. So

Type $123\sim31+a_{1}$ in Desmos. Here $a_{1}$ stands for $a$, and $\sim$ tells Desmos to find the value that matches both sides. Under PARAMETERS, Desmos shows $a_{1}=92$, so the whole angle at A measures $92^\circ$.

The angle bisector $AD$ makes $\angle BAD$ half of $\angle BAC$, so $m\angle BAD=\frac{a_{1}}{2}$. Type $\frac{a_{1}}{2}$ beneath the first line in Desmos. It prints $46$, the angle at $A$ inside triangle $ABD$.

The three angles inside triangle $ABD$ are $46^\circ$ at $A$, $31^\circ$ at $B$, and $x^\circ$ at $D$. A triangle's angles add to $180^\circ$, so

Type $46+31+x_{1}\sim180$ in Desmos, using $x_{1}$ for the requested angle $x$. Under PARAMETERS, it shows $x_{1}=103$. So $\angle ADB$ measures $103^\circ$. Grid in 103.