https://file.aiquickdraw.com/mathbot/file/17264940195323q78nnom.jpg

Let's analyze the provided image and write out the reasoning for the two-column proof.

### Given:
1. \( m \angle ABC = 90^\circ \)
2. \( \overline{BD} \) bisects \( \angle ABC \)

### To Prove:
\( m \angle DBC = 45^\circ \)

### Proof:

| **Statements**                               | **Reasons**                                             |
|---------------------------------------------|---------------------------------------------------------|
| 1. \( m \angle ABC = 90^\circ \)            | Given                                                   |
| 2. \( \overline{BD} \) bisects \( \angle ABC \) | Given                                                   |
| 3. \( \angle ABD = \angle DBC \)            | Definition of angle bisector                            |
| 4. \( m \angle ABC = m \angle ABD + m \angle DBC \) | Angle Addition Postulate                                |
| 5. \( m \angle ABD + m \angle DBC = m \angle ABC \) | Substitution (from statement 4)                         |
| 6. \( m \angle DBC + m \angle DBC = m \angle ABC \) | Substitution (since \( \angle ABD = \angle DBC \))       |
| 7. \( 2(m \angle DBC) = m \angle ABC \)     | Combine like terms                                      |
| 8. \( 2(m \angle DBC) = 90^\circ \)         | Substitution (from statement 1)                         |
| 9. \( m \angle DBC = 45^\circ \)            | Solve equation \( 2(m \angle DBC) = 90^\circ \)         |

### Explanation:

- **Step 1**: The problem gives the measure of angle \( \angle ABC = 90^\circ \).
- **Step 2**: It is given that \( \overline{BD} \) bisects \( \angle ABC \), meaning it divides the angle into two equal parts.
- **Step 3**: Since \( \overline{BD} \) is an angle bisector, we conclude that \( \angle ABD = \angle DBC \).
- **Step 4-5**: Using the angle addition postulate, we express \( m \angle ABC \) as the sum of the two smaller angles, \( m \angle ABD + m \angle DBC \).
- **Step 6**: Substituting \( m \angle ABD = m \angle DBC \) (since they are equal), we get \( m \angle ABC = 2(m \angle DBC) \).
- **Step 7-8**: Solving for \( m \angle DBC \), we know \( m \angle ABC = 90^\circ \), so we divide by 2 to get \( m \angle DBC = 45^\circ \).

Would you like further clarification or details on any step?

### Relative Questions:
1. What is the angle addition postulate, and how is it applied in this proof?
2. Why is \( \angle ABD = \angle DBC \) when \( \overline{BD} \) bisects \( \angle ABC \)?
3. How do we derive the equation \( 2(m \angle DBC) = m \angle ABC \)?
4. How would the proof change if \( m \angle ABC \) was different from \( 90^\circ \)?
5. What is the importance of bisectors in geometry proofs like this?

### Tip:
In geometry proofs, clearly defining your given information and applying definitions (such as bisectors or angle relationships) early in the proof helps structure the logical steps and ensure clarity throughout.

Prove that m ∠DBC = 45° in this geometry problem involving angle bisectors and the angle addition postulate. This step-by-step solution covers important concepts like bisectors and angle addition, suitable for Grades 8-10 students.

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