https://file.aiquickdraw.com/mathbot/file/1729823400496bsmlziga.jpg

The problem asks to solve for the variable \( x \) and find the measures of various angles, given that \( \overrightarrow{GH} \) bisects \( \angle FGI \). From the image, we see the following information:

- \( m\angle FGH = 4x - 6^\circ \)
- \( m\angle HGI = 5x - 14^\circ \)
- Since \( GH \) bisects \( \angle FGI \), we know that:
  \[
  m\angle FGH = m\angle HGI
  \]

### Step 1: Solve for \( x \)
We can set the two angle expressions equal to each other:
\[
4x - 6 = 5x - 14
\]

Now, solve for \( x \):
\[
4x - 5x = -14 + 6
\]
\[
-x = -8
\]
\[
x = 8
\]

### Step 2: Find \( m\angle FGH \)
Substitute \( x = 8 \) into the expression for \( m\angle FGH \):
\[
m\angle FGH = 4(8) - 6 = 32 - 6 = 26^\circ
\]

### Step 3: Find \( m\angle HGI \)
Since \( GH \) bisects the angle, \( m\angle HGI = m\angle FGH = 26^\circ \).

### Step 4: Find \( m\angle FGI \)
Since \( GH \) bisects \( \angle FGI \), the total angle \( \angle FGI \) is twice the measure of \( \angle FGH \):
\[
m\angle FGI = 2 \times 26^\circ = 52^\circ
\]

### Final Answers:
- \( x = 8 \)
- \( m\angle FGH = 26^\circ \)
- \( m\angle HGI = 26^\circ \)
- \( m\angle FGI = 52^\circ \)

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the relationship between the bisector and the angles it splits?
2. How would the solution change if \( GH \) did not bisect \( \angle FGI \)?
3. Can you generalize the approach to solving for bisected angles in terms of algebraic expressions?
4. What if the angle expressions were non-linear—how would the process of solving change?
5. How does angle bisecting apply in triangle geometry?

**Tip:** Always check if the angles form a bisected pair, as it simplifies finding unknown variables!

In the diagram, GH bisects ∠FGI. Solve for x and find m∠FGH. Find m∠HGI. Find m∠FGI.

Learn how to solve for x and find angle measures in a geometry problem where an angle bisector is involved. Using algebra and the Angle Bisector Theorem, this step-by-step solution is designed for students in Grades 7-10.

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