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!