In the provided diagram, the line segment GH bisects the angle ∠FGL. The two angles formed are expressed as:

• ∠FGH = $(4x - 5)^\circ$
• ∠HGI = $(3x - 15)^\circ$

Since GH bisects ∠FGL, these two angles should be equal. We can set up an equation to solve for $x$:

$4x - 5 = 3x - 15$

Step 1: Solve for $x$

1. Subtract $3x$ from both sides: $x - 5 = -15$

2. Add 5 to both sides: $x = -15 + 5 = -10$

So, $x = 10$.

Step 2: Find m∠FGH

Substitute $x = 10$ into the expression for m∠FGH:

$m∠FGH = 4x - 5 = 4(10) - 5 = 40 - 5 = 35^\circ$

Since GH bisects ∠FGL, m∠HGI is also 35°.

Step 3: Find m∠FGL

Since ∠FGL consists of both ∠FGH and ∠HGI, we add these angles together:

$m∠FGL = m∠FGH + m∠HGI = 35^\circ + 35^\circ = 70^\circ$

Final answers:

• $x = 10$
• $m∠FGH = 35^\circ$
• $m∠HGI = 35^\circ$
• $m∠FGL = 70^\circ$

Would you like more details or have any questions? Here are 5 related questions you could explore:

1. How do we use the concept of angle bisectors in real-world applications?
2. What are alternate ways to verify if two angles are equal?
3. How do properties of angles change with different bisectors?
4. Can you explain how to solve multi-variable angle problems using algebra?
5. How does changing the value of $x$ affect the total angle in this scenario?

Tip: Always check for angle relationships such as supplementary, complementary, or bisected angles to simplify geometric problems.