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!