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Let's analyze the problem and solve it step-by-step.

### Problem breakdown:

- \( \overrightarrow{BF} \) bisects \( \angle DBG \), which means that \( \angle DBF = \angle FBG \).
- We are given the following expressions:
  - \( m\angle DBF = 8x - 10 \)
  - \( m\angle FBG = 5x + 23 \)

Since \( BF \) bisects \( \angle DBG \), we can set these two angle measures equal to each other:

\[
8x - 10 = 5x + 23
\]

### Step 1: Solve for \( x \)

Let's solve this equation for \( x \).

\[
8x - 5x = 23 + 10
\]
\[
3x = 33
\]
\[
x = 11
\]

### Step 2: Calculate the measures of \( \angle DBF \) and \( \angle FBG \)

Now that we have \( x = 11 \), substitute it back into the expressions for \( m\angle DBF \) and \( m\angle FBG \).

- \( m\angle DBF = 8x - 10 = 8(11) - 10 = 88 - 10 = 78^\circ \)
- \( m\angle FBG = 5x + 23 = 5(11) + 23 = 55 + 23 = 78^\circ \)

Since \( BF \) bisects \( \angle DBG \), the full angle \( \angle DBG = 2 \times 78^\circ = 156^\circ \).

### Step 3: Calculate the remaining angles

- \( \angle DBE \) and \( \angle ABF \) appear to form a linear pair (180° total) with \( \angle DBG \).
- Since \( \angle DBG = 156^\circ \), the remaining part of the straight line will be \( \angle CBE = 180^\circ - 156^\circ = 24^\circ \).
- \( \angle ABE = 180^\circ \) because it's a straight line.

Thus, the angles are:

- \( m\angle DBF = 78^\circ \)
- \( m\angle FBG = 78^\circ \)
- \( m\angle DBG = 156^\circ \)
- \( m\angle CBE = 24^\circ \)
- \( m\angle ABE = 180^\circ \)
- \( m\angle ABF = 180^\circ \)

### Summary of answers:

- \( x = 11 \)
- \( m\angle DBF = 78^\circ \)
- \( m\angle FBG = 78^\circ \)
- \( m\angle DBG = 156^\circ \)
- \( m\angle CBE = 24^\circ \)
- \( m\angle ABE = 180^\circ \)
- \( m\angle ABF = 180^\circ \)

Would you like further details or explanations? Here are some related questions:

1. How does the property of angle bisection help in geometry problems?
2. What is the relationship between a linear pair of angles?
3. Can you explain how to find unknown angles in polygons?
4. What is the significance of supplementary angles in this problem?
5. How can you verify if two angles are equal when given algebraic expressions?

**Tip**: When solving for angles, always check if the given lines are straight or form supplementary angles.

If BF bisects ∠DBG, m∠DBF = 8x - 10 and m∠FBG = 5x + 23, find each missing measure.

Learn how to solve for missing angles when given algebraic expressions for angles bisected by a ray. This problem involves solving for x and applying the linear pair theorem to find the remaining angle measures.

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