https://file.aiquickdraw.com/mathbot/file/1726064759890ru5q6g12.jpg

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We are given:
- \( m\angle ABD = 78^\circ \)
- \( m\angle ABC = 46^\circ \)
- The expressions for angles:
  - \( \angle DBC = (7x - 3)^\circ \)
  - \( \angle DAB = (4x + 4)^\circ \)

We need to find \( m\angle DBC \).

### Step-by-step Solution:

1. **Find \( m\angle DBC \) using given angles:**

   Since \( m\angle ABD = m\angle ABC + m\angle DBC \), we can write:

   \[
   78^\circ = 46^\circ + m\angle DBC
   \]

2. **Solve for \( m\angle DBC \):**

   \[
   m\angle DBC = 78^\circ - 46^\circ = 32^\circ
   \]

Thus, \( m\angle DBC = 32^\circ \).

Would you like more details or have any questions? Here are some related questions you might explore:

1. What is the measure of \( \angle DAB \) using the expressions given?
2. How do you solve for \( x \) using the expressions for \( \angle DAB \) and \( \angle DBC \)?
3. What is the relationship between angles on a straight line?
4. How can you verify if the given values are correct?
5. How do supplementary angles relate to this problem?

**Tip:** When solving for unknowns, always check if there are multiple equations you can use to verify your solution.

If m∠ABD = 78°, what are m∠ABC and m∠DBC?

Learn how to calculate the measures of angles m∠ABC and m∠DBC in a triangle given m∠ABD = 78°. This problem explores the supplementary angles theorem and is suitable for students in Grades 6-8.

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