Let's break down the problem step by step from the image you've provided.

Information from the Diagram:

• The diagram shows that $GH$ bisects $\angle FGI$.
• The angle $\angle FGH$ is labeled as $(3x - 3)^\circ$.
• The angle $\angle HGI$ is labeled as $(4x + 17)^\circ$.
• Since $GH$ bisects $\angle FGI$, this means $\angle FGH = \angle HGI$.

Part A: Solve for $x$ and Find $m \angle FGH$

We know that since $GH$ bisects the angle $FGI$, the two smaller angles, $\angle FGH$ and $\angle HGI$, are equal:

$\angle FGH = \angle HGI$

This gives the equation:

$3x - 3 = 4x + 17$

Now, solving for $x$:

$3x - 4x = 17 + 3$

$-x = 20$

$x = -20$

Now that we know $x = -20$, substitute this value back into $\angle FGH$ to find the measure of that angle:

$m \angle FGH = 3(-20) - 3 = -60 - 3 = -63^\circ$

Part B: Find $m \angle HGI$

Since $GH$ bisects $\angle FGI$, we already know that $m \angle HGI = m \angle FGH$. Therefore:

$m \angle HGI = -63^\circ$

Part C: Find $m \angle FGI$

Since $\angle FGH$ and $\angle HGI$ are equal and together make up $\angle FGI$:

$m \angle FGI = m \angle FGH + m \angle HGI$

$m \angle FGI = -63^\circ + (-63^\circ) = -126^\circ$

Summary:

• $x = -20$
• $m \angle FGH = -63^\circ$
• $m \angle HGI = -63^\circ$
• $m \angle FGI = -126^\circ$

Do you want further details on any part of the solution?

Here are 5 related questions to expand on this topic:

1. What happens if the bisector is not given? How do you solve for the angles?
2. How would you find the missing angle if this was a triangle?
3. Can you explain how angle bisectors work in polygons?
4. What is the sum of interior angles for different polygons, and how is it useful?
5. What other geometric properties involve angle bisectors?

Tip: When solving for unknown angles, always check if angles are equal, supplementary, or complementary to simplify the problem.