To solve the problem, we need to find the measure of $\angle ABC$.

Given:

• $m \angle ABD = 104^\circ$
• $m \angle EBD = 33^\circ$
• $\overline{BE}$ bisects $\angle CBD$

Step-by-Step Solution:

1. Interpretation of bisector property: Since $\overline{BE}$ bisects $\angle CBD$, the measure of $\angle CBE$ is equal to the measure of $\angle EBD$. Therefore: $m \angle CBE = m \angle EBD = 33^\circ$

2. Relationship between the angles:

   • The full angle $\angle ABD$ is made up of $\angle ABC$ and $\angle CBD$.
   • $\angle CBD$ consists of two equal parts: $\angle CBE$ and $\angle EBD$.

   Therefore, we can write: $m \angle ABD = m \angle ABC + m \angle CBE + m \angle EBD$

3. Substitute the known values: $104^\circ = m \angle ABC + 33^\circ + 33^\circ$

4. Solve for $m \angle ABC$: $104^\circ = m \angle ABC + 66^\circ$ $m \angle ABC = 104^\circ - 66^\circ = 38^\circ$

Thus, the measure of $\angle ABC$ is $\boxed{38^\circ}$.

Would you like more details or explanations on this problem? Here are some related questions to explore:

1. How do you calculate angles when a bisector is involved in different geometric setups?
2. What is the general formula for finding unknown angles in a triangle?
3. How does the Angle Addition Postulate apply to complex angle problems like this?
4. Can we use the Law of Cosines for non-right triangles to verify angle measures?
5. How do bisectors affect the properties of triangles in geometry?

Tip: Always break down geometric figures into smaller, simpler parts when working with bisectors and angle sums.