The image shows a geometry problem where BD bisects angle ∠ABC, and we are asked to find the measure of ∠ABC. The given angles are:

• ∠ABD = $(6x + 16)^\circ$
• ∠DBC = $(8x - 23)^\circ$

Since BD bisects ∠ABC, it means that:

$\text{∠ABD} = \text{∠DBC}$

So, we can set up the equation:

$6x + 16 = 8x - 23$

Let's solve for $x$.The value of $x$ is $\frac{39}{2}$ or 19.5.

Now, let's use this value of $x$ to calculate the measures of ∠ABD and ∠DBC, and then find the full measure of ∠ABC by doubling one of these angles (since ∠ABD = ∠DBC).