This problem involves using the basic principles of similar triangles or proportionality between segments formed by parallel lines intersected by a transversal.

Observations:

1. $GK \parallel HL \parallel JM$: The given lines are parallel.
2. The segments on the transversal are divided proportionally.

Given:

• $GH = 8$,
• $KL = 10$,
• $HJ = 6$,
• $LM = x$.

From the parallel lines, we know that: $\frac{GH}{HJ} = \frac{KL}{LM}.$

Substituting the known values:

$\frac{8}{6} = \frac{10}{x}.$

Solving for $x$:

1. Cross multiply: $8x = 6 \cdot 10.$
2. Simplify: $8x = 60.$
3. Divide by 8: $x = \frac{60}{8} = 7.5.$

Final Answer:

$x = 7.5.$

Would you like me to explain the reasoning further or clarify any specific step? 😊

5 Related Questions:

1. What are the conditions for applying the proportionality rule in triangles?
2. How do you prove that the triangles formed here are similar?
3. What happens if one of the parallel lines is missing—how do proportions change?
4. How can this proportionality be extended to real-world applications like maps?
5. How do you derive the formula for proportionality between segments in parallel lines?

Tip:

When working with parallel lines and a transversal, always look for opportunities to use proportionality or similar triangles—it’s a powerful tool for solving geometry problems!