If m∠CBE = 4x - 9 and m∠FBE = 5x + 15, find m∠CBE.  Round to the nearest tenths place.*


To find \( m \angle CBE \), we first need to determine the value of \(x\) using the given information.

### Given:
- \( m \angle CBE = 4x - 9 \)
- \( m \angle FBE = 5x + 15 \)

### Step-by-Step Solution:

1. **Understand the Angle Relationship:**
   - \( \angle CBE \) and \( \angle FBE \) form a linear pair because they are adjacent and lie on a straight line. Thus, these angles are supplementary, meaning their sum is \(180^\circ\).

2. **Set Up the Equation:**
   - Since \( \angle CBE + \angle FBE = 180^\circ \), we have:
   \[
   (4x - 9) + (5x + 15) = 180
   \]

3. **Combine Like Terms:**
   \[
   4x + 5x - 9 + 15 = 180
   \]
   \[
   9x + 6 = 180
   \]

4. **Solve for \(x\):**
   \[
   9x = 180 - 6
   \]
   \[
   9x = 174
   \]
   \[
   x = \frac{174}{9}
   \]
   \[
   x = 19.3333\ldots \text{ (repeating)}
   \]

5. **Find \( m \angle CBE \):**
   - Substitute \(x = 19.3333\ldots\) into the expression for \( m \angle CBE \):
   \[
   m \angle CBE = 4x - 9
   \]
   \[
   m \angle CBE = 4(19.3333\ldots) - 9
   \]
   \[
   m \angle CBE = 77.3333\ldots - 9
   \]
   \[
   m \angle CBE = 68.3333\ldots
   \]

6. **Round to the Nearest Tenth:**
   \[
   m \angle CBE \approx 68.3
   \]

### Conclusion:
The value of \( m \angle CBE \) rounded to the nearest tenth place is **68.3**.

Would you like more details on the solution, or do you have any questions? 

Here are some related questions you might find interesting:
1. How do you solve for \(x\) in equations involving linear pairs?
2. What is the importance of supplementary angles in geometry?
3. Can you find \( m \angle FBE \) after determining \(x\)?
4. How do different angle relationships (like complementary and vertical) affect the solution?
5. How would the solution change if the angles were not supplementary?

**Tip:** Always check the angle relationships in geometry problems to determine how to set up your equations correctly.

Learn how to find the measure of angle CBE in a geometry problem involving supplementary angles. Follow a step-by-step solution to determine the value of angle CBE using given angle measures. Suitable for high school students studying geometry.

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