Problem Analysis

Given that $\overrightarrow{GH}$ bisects $\angle FGI$, we need to solve the following:

a. Solve for $x$ and find $m\angle FGH$.
b. Find $m\angle HGI$.
c. Find $m\angle FGI$.

From the diagram:

• $\angle FGH = (2x - 1)^\circ$
• $\angle HGI = (3x - 19)^\circ$

Since $\overrightarrow{GH}$ bisects $\angle FGI$, we know:

$m\angle FGH = m\angle HGI$

Step 1: Solve for $x$

Set the expressions for $\angle FGH$ and $\angle HGI$ equal to each other:

$2x - 1 = 3x - 19$

Simplify the equation:

$-1 + 19 = 3x - 2x$

$18 = x$

Step 2: Find $m\angle FGH$

Substituting $x = 18$ into $\angle FGH$:

$m\angle FGH = 2(18) - 1 = 36 - 1 = 35^\circ$

Step 3: Find $m\angle HGI$

Since $m\angle HGI = m\angle FGH$, we have:

$m\angle HGI = 35^\circ$

Step 4: Find $m\angle FGI$

Since $\angle FGI$ is bisected by $\overrightarrow{GH}$, the total measure of $\angle FGI$ is:

$m\angle FGI = m\angle FGH + m\angle HGI = 35^\circ + 35^\circ = 70^\circ$

Final Answers:

a. $x = 18$ and $m\angle FGH = 35^\circ$.
b. $m\angle HGI = 35^\circ$.
c. $m\angle FGI = 70^\circ$.

Would you like further clarification on any part of this solution? Here are some related questions:

1. How do you identify an angle bisector in geometric diagrams?
2. What are some common strategies to solve equations involving angle measures?
3. How do you determine the total angle measure when given a bisected angle?
4. Can you apply the concept of angle bisectors to solve other types of geometry problems?
5. What are the properties of angle bisectors in different types of triangles?

Tip: Always double-check your substitutions when solving equations to ensure accurate results!