1. The problem asks to find the measure of $\angle ABC$ given that $BD$ is an angle bisector in triangle $ABC$ with $\angle ABD = x + 6$ and $\angle DBC = 2x - 3$. 2. Since $BD$ is an angle bisector, it divides $\angle ABC$ into two equal angles. Therefore, we set: $$x + 6 = 2x - 3$$ 3. Solve for $x$: $$x + 6 = 2x - 3$$ $$6 + 3 = 2x - x$$ $$9 = x$$ 4. Substitute $x = 9$ back into one of the angle expressions to find $\angle ABD$: $$\angle ABD = x + 6 = 9 + 6 = 15^\circ$$ 5. Since $BD$ bisects $\angle ABC$, the full angle is twice $15^\circ$: $$\angle ABC = 2 \times 15 = 30^\circ$$ Final answer: $$\boxed{30^\circ}$$