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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.

En la figura siguiente, m∠ABD = 104°, m∠EBD = 33°, y BE biseca ∠CBD. Hallar m∠ABC.

Learn how to find the measure of ∠ABC when given angles m∠ABD = 104° and m∠EBD = 33°, with BE bisecting ∠CBD. This problem uses key geometry concepts like angle bisectors and the Angle Addition Postulate. Suitable for students in Grades 8-10.

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