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The examples and diagram you've provided focus on solving angle bisectors and equations for angles.

Here’s a breakdown of the examples based on the images:

### Example 3:
Given:
- \( m \angle DEG = 88^\circ \)
- \( EF \) bisects \( \angle DEG \), which means \( \angle FEG \) is half of \( \angle DEG \).

Thus, \( m \angle FEG = \frac{1}{2} \times 88^\circ = 44^\circ \).

The provided answer \( 44^\circ \) is correct.

### Example 4:
Given:
- \( m \angle FED = 27^\circ \)
- \( EF \) bisects \( \angle DEG \), and we need to find \( m \angle GED \).

Since \( EF \) is the bisector, both \( m \angle FED \) and \( m \angle GED \) are equal.

So, \( m \angle GED = 27^\circ \).

The answer \( 54^\circ \) written in the example likely refers to the sum of the two angles ( \( m \angle FED + m \angle GED \)).

### Example 5:
Given:
- \( m \angle DEF = 3x + 1 \)
- \( m \angle DEG = 5x + 19 \)

Since \( EF \) bisects \( \angle DEG \), we have:
\[ m \angle DEF + m \angle FEG = m \angle DEG \]
But because \( EF \) bisects the angle, the two parts should be equal. We can write:
\[ 3x + 1 + 3x + 1 = 5x + 19 \]
Simplifying:
\[ 6x + 2 = 5x + 19 \]
Subtract \( 5x \) from both sides:
\[ x + 2 = 19 \]
Subtract 2 from both sides:
\[ x = 17 \]

Thus, \( x = 17 \).

### Example 6:
Given:
- \( m \angle DEF = 5x - 3 \)
- \( m \angle FEG = 2x + 15 \)

Since \( EF \) bisects \( \angle DEG \), the two angles are equal:
\[ 5x - 3 = 2x + 15 \]
Solving for \( x \):
Subtract \( 2x \) from both sides:
\[ 3x - 3 = 15 \]
Add 3 to both sides:
\[ 3x = 18 \]
Divide by 3:
\[ x = 6 \]

Thus, \( x = 6 \).

### Example 7:
Given:
- \( m \angle FEG = 6x - 7 \)
- \( m \angle FED = 2x + 41 \)

We need to solve for \( x \) first.

Since \( EF \) bisects \( \angle DEG \), the two angles are equal:
\[ 6x - 7 = 2x + 41 \]
Solving for \( x \):
Subtract \( 2x \) from both sides:
\[ 4x - 7 = 41 \]
Add 7 to both sides:
\[ 4x = 48 \]
Divide by 4:
\[ x = 12 \]

Now, to find \( m \angle DEG \), sum the two angles:
\[ m \angle FEG + m \angle FED = (6x - 7) + (2x + 41) \]
Substitute \( x = 12 \):
\[ (6(12) - 7) + (2(12) + 41) = (72 - 7) + (24 + 41) = 65 + 65 = 130^\circ \]

Thus, \( m \angle DEG = 130^\circ \).

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Would you like further details on any of these solutions?

### 5 Follow-up Questions:
1. How is an angle bisector used to split an angle into two equal parts?
2. Can you explain how solving for \( x \) helps in finding the value of an angle?
3. What role does substitution play in checking the final angle?
4. How does combining like terms simplify solving equations?
5. Why is it important to verify the values of all angles in a bisected diagram?

### Tip:
Always check your work by plugging the value of \( x \) back into the original expressions to ensure accuracy.

Solving for angles based on bisectors and algebraic equations for different examples involving bisected angles.

Learn how to solve algebraic equations involving bisected angles like ∠DEG using angle bisector principles and linear equations. Detailed step-by-step solutions provided for various examples. Suitable for Grades 8-10.

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