https://file.aiquickdraw.com/mathbot/file/1727924846309cfc5d2g9.jpg

Let's analyze the provided diagram and solve the problem step by step.

### Information from the diagram:
- We are given that \( GH \) bisects \( \angle FGI \), which means that the two smaller angles on either side of \( GH \) are equal.
- The two angles are labeled as \( (2x - 3)^\circ \) and \( (3x - 20)^\circ \).

### Part (a) Solve for \( x \) and find \( m\angle FGH \):
Since \( GH \) bisects \( \angle FGI \), we can set the two angles equal to each other:
\[
2x - 3 = 3x - 20
\]
Solve for \( x \):
\[
2x - 3 = 3x - 20
\]
Subtract \( 2x \) from both sides:
\[
-3 = x - 20
\]
Add 20 to both sides:
\[
x = 17
\]

Now that we have \( x = 17 \), substitute this into one of the angle expressions (say, \( \angle FGH = 2x - 3 \)) to find \( m\angle FGH \):
\[
m\angle FGH = 2(17) - 3 = 34 - 3 = 31^\circ
\]

### Part (b) Find \( m\angle HGI \):
Since \( GH \) bisects \( \angle FGI \), we know that \( \angle FGH = \angle HGI \), so:
\[
m\angle HGI = 31^\circ
\]

### Part (c) Find \( m\angle FGI \):
The total angle \( \angle FGI \) is the sum of the two smaller angles:
\[
m\angle FGI = m\angle FGH + m\angle HGI = 31^\circ + 31^\circ = 62^\circ
\]

### Final Answers:
- \( x = 17 \)
- \( m\angle FGH = 31^\circ \)
- \( m\angle HGI = 31^\circ \)
- \( m\angle FGI = 62^\circ \)

---

Let me know if you want further clarification! Here are 5 related questions to explore this topic further:
1. What is the definition of an angle bisector in geometry?
2. If \( GH \) bisects \( \angle FGI \), what other properties can we infer about line \( GH \)?
3. How do we solve linear equations involving angles in other types of problems?
4. What if the two angles given were not equal—how would that change the problem?
5. How does the concept of angle bisectors apply in triangle geometry?

**Tip:** Always check your calculations by verifying if your results satisfy the original problem conditions, such as equal angles when dealing with bisectors.

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

In this problem, learn how to solve for x in an angle bisector diagram where GH bisects ∠FGI, and ∠FGH = (2x - 3)° and ∠HGI = (3x - 20)°. Includes step-by-step algebraic solution and theorems.

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