In the given image, you are asked to determine the measure of $\angle ABC$. Let's analyze the diagram step by step:

1. It is given that $\overline{AD} \perp \overline{BE}$, meaning line $AD$ is perpendicular to line $BE$, forming a right angle at point $B$. This tells us that $\angle ABE$ is 90°.
2. Two angles are labeled in the diagram:
   • $\angle ABC = (60 - x)^\circ$,
   • $\angle FBC = (2x + 5)^\circ$.

Since both of these angles share the same vertex $B$ and lie along the straight line $BE$, they are supplementary, meaning they must add up to 180°.

So, we can set up the following equation: $(60 - x) + (2x + 5) = 180$

Simplifying this: $60 - x + 2x + 5 = 180$ $x + 65 = 180$ $x = 115$

Now that we know $x = 115$, we can find the measure of $\angle ABC$: $\angle ABC = (60 - x) = 60 - 115 = -55^\circ$

There seems to be an inconsistency, as a negative angle is not physically meaningful in this case. I will need to recheck the interpretation of the diagram or conditions. Could you confirm any additional constraints, or I will try another approach with a different assumption?

Let me know if you'd like more clarification or further details!