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In this geometry problem, we are working with the measures of angles \( \angle ABC \), \( \angle ABD \), and \( \angle DBC \), and the steps involve solving for \( x \) and calculating the measures of these angles. Based on what I see, the steps are labeled with specific properties or postulates.

Let’s walk through the problem and determine which property goes into each box.

1. **Given Information**:
   - \( m \angle ABC = (8x)^\circ \)
   - \( m \angle ABD = (3x + 5)^\circ \)
   - \( m \angle DBC = (6x - 16)^\circ \)

2. **Equation Set-up**:
   - \( m \angle ABC = m \angle ABD + m \angle DBC \)
     - This is the **Angle Addition Postulate**.

3. **Substitution** of the angle values into the equation:
   - \( 8x = (3x + 5) + (6x - 16) \)

4. **Combine like terms** on the right side:
   - \( 8x = 9x - 11 \)
   - This follows from the **Addition Property of Equality** (simplifying both sides of the equation).

5. **Isolate \( x \)** by subtracting \( 9x \) from both sides:
   - \( -x = -11 \)
   - This step applies the **Subtraction Property of Equality**.

6. **Solve for \( x \)** by dividing both sides by -1:
   - \( x = 11 \)
   - This uses the **Division Property of Equality**.

7. **Substitute** \( x = 11 \) back into \( m \angle ABD \):
   - \( m \angle ABD = 3(11) + 5 \)
     - This is a **Substitution** step.

8. **Simplify**:
   - \( m \angle ABD = 33 + 5 = 38^\circ \)
   - This final step is **Simplification**.

### Now, based on these steps, the boxes are filled as follows:

1. **Angle Addition Postulate**: Goes next to \( m \angle ABC = m \angle ABD + m \angle DBC \).
2. **Addition Property of Equality**: Goes next to the equation \( 8x = (3x + 5) + (6x - 16) \).
3. **Subtraction Property of Equality**: Goes next to \( -x = -11 \).
4. **Substitution**: Used for the step where \( x = 11 \) is substituted into \( m \angle ABD = 3(11) + 5 \).
5. **Simplification**: Used for the step where \( m \angle ABD = 38^\circ \).

Would you like further details or have any questions?

Here are some related questions to expand on this:
1. What is the difference between the Angle Addition Postulate and Segment Addition Postulate?
2. How do you solve for unknowns in similar angle problems?
3. What is the Substitution Property in geometry, and how is it different from algebra?
4. How does combining like terms help in solving equations?
5. Can angle measures be negative? Why or why not?

**Tip**: When solving equations involving angles, always ensure that like terms (those with \( x \) or constants) are properly combined to simplify the process.

Learn how to solve geometry problems by calculating angle measurements and solving for unknown variables. This problem involves understanding and applying several properties and postulates, including the Angle Addition Postulate and various properties of equality. Suitable for middle school to high school students.

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