https://file.aiquickdraw.com/mathbot/file/1726779684056lppw1j10.jpg

Let's break down the problem based on the image you provided.

Given:
- Line **GH** bisects ∠FGI.
- The angle measurements are given as:
  - ∠FGH = \( (5x - 4)^\circ \)
  - ∠HGI = \( (6x - 11)^\circ \)

### Step 1: Solve for **x** (Part a)
Since **GH** bisects ∠FGI, it means that the two angles ∠FGH and ∠HGI are equal. Therefore, we can set up the equation:

\[
5x - 4 = 6x - 11
\]

Solving for **x**:
\[
5x - 4 = 6x - 11
\]
\[
-4 + 11 = 6x - 5x
\]
\[
7 = x
\]

So, **x = 7**.

### Step 2: Find ∠FGH (Part a)
Now that we have **x = 7**, substitute this into the expression for ∠FGH:

\[
\text{m}∠FGH = 5x - 4 = 5(7) - 4 = 35 - 4 = 31^\circ
\]

### Step 3: Find ∠HGI (Part b)
Substitute **x = 7** into the expression for ∠HGI:

\[
\text{m}∠HGI = 6x - 11 = 6(7) - 11 = 42 - 11 = 31^\circ
\]

### Step 4: Find ∠FGI (Part c)
Since ∠FGI is the full angle, and **GH** bisects ∠FGI into two equal parts, the total measure of ∠FGI is:

\[
\text{m}∠FGI = 2 \times \text{m}∠FGH = 2 \times 31^\circ = 62^\circ
\]

### Final Answers:
a. \( x = 7 \), \( \text{m}∠FGH = 31^\circ \)  
b. \( \text{m}∠HGI = 31^\circ \)  
c. \( \text{m}∠FGI = 62^\circ \)

---

Would you like more details or clarification on any step? Here are 5 additional related questions you can explore:

1. How do we determine the total measure of a bisected angle in general?
2. Can you find another method to check if the value of **x** is correct?
3. What is the geometric significance of a bisected angle in terms of symmetry?
4. How would the solution change if **GH** did not bisect the angle?
5. Can this problem be solved using trigonometric properties?

**Tip:** Always verify your angle relations when working with bisected angles to ensure accuracy in geometry problems!

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

Learn how to solve for x and find the measures of angles in a bisected geometry problem. This solution uses algebra and geometry to determine the values of m∠FGH, m∠HGI, and m∠FGI. Suitable for students in Grades 7-9.

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