We are given that $BD$ bisects $\angle ABC$, and we need to find the value of $x$.

Steps to solve:

1. The diagram shows that $BD$ bisects $\angle ABC$. Therefore, $\angle ABD = \angle DBC$.
2. The full measure of $\angle ABC$ is given as $53^\circ$.
3. Since $BD$ bisects $\angle ABC$, each of the angles $\angle ABD$ and $\angle DBC$ will be half of $53^\circ$: $x = \frac{53^\circ}{2} = 26.5^\circ$

So, the value of $x$ is $26.5^\circ$.

Would you like more details or have any questions?

Related questions:

1. How would the answer change if $\angle ABC$ was a different angle?
2. What is the significance of angle bisectors in geometry?
3. Can angle bisectors divide angles into unequal parts in any geometric figures?
4. How can we apply angle bisectors to find unknown sides in triangles?
5. How do we use trigonometry with bisected angles?

Tip: In any angle bisection, remember that the bisector divides the angle into two equal parts, making it a useful tool for geometric calculations.