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In the provided image, you are asked to find the measure of angle \( \angle LHB \) given certain information about the angles in the diagram.

### Problem Breakdown:
1. **Given:**
   - \( \overrightarrow{BA} \) and \( \overrightarrow{BC} \) are opposite rays.
   - \( \overrightarrow{BE} \) bisects \( \angle EBC \).
   - \( \overrightarrow{BF} \) bisects \( \angle ABF \).
   - \( m\angle EBC = (31x - 21)^\circ \).
   - \( m\angle EBH = (4x + 45)^\circ \).

2. **To Find:**
   - The measure of \( \angle LHB \).

### Steps to Solve:

1. **Understanding the Geometry:**
   - Since \( \overrightarrow{BE} \) is the bisector of \( \angle EBC \), we have \( \angle EBA = \angle EBC \).
   - Similarly, \( \overrightarrow{BF} \) bisects \( \angle ABF \).

2. **Angle Relationships:**
   - Since \( \overrightarrow{BA} \) and \( \overrightarrow{BC} \) are opposite rays, the entire angle \( \angle ABC \) should sum to \( 180^\circ \).
   - Therefore, \( m\angle EBC + m\angle ABF + m\angle HBC = 180^\circ \).

3. **Using the given angles:**
   - \( m\angle EBC = (31x - 21)^\circ \).
   - \( m\angle EBH = (4x + 45)^\circ \) (this angle is part of \( \angle EBC \)).
   - The required \( m\angle LHB = m\angle EBH + m\angle HBC \).

4. **Equation Setup:**
   - The sum of \( m\angle EBC \) and \( m\angle ABF \) forms part of the equation for \( m\angle ABC = 180^\circ \).

   Since \( \angle LHB \) forms part of the bisected angle, we can directly calculate it by substituting the values given.

5. **Substitute and Solve:**

   \[
   m\angle LHB = (4x + 45)^\circ + m\angle HBC
   \]

   The key is first to solve for \( x \) using the relationships provided.

6. **Find \( x \):**
   \[
   m\angle EBC = 2 \times m\angle EBH
   \]
   \[
   (31x - 21)^\circ = 2(4x + 45)^\circ
   \]
   \[
   31x - 21 = 8x + 90
   \]
   \[
   23x = 111
   \]
   \[
   x = \frac{111}{23}
   \]
   Simplify to find \( x \).

Once \( x \) is found, substitute back to find \( m\angle LHB \).

Would you like to continue with solving \( x \) and finding the final angle?

In the figure, BA and BC are opposite rays. BE bisects ∠EBC and BF bisects ∠ABF. If m∠EBC = (31x - 21)° and m∠EBH = (4x + 45)°, find m∠LHB.

This problem involves finding the measure of angle LHB using the properties of angle bisectors and the concept of linear pair of angles. Suitable for students in Grades 10-12, it explores how to use given angle expressions and solve for unknowns.

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