https://r2.erweima.ai/mathbot/image/1725545722904liboo9hy.jpg

In the given image, we see a set of parallel lines \(\mathcal{L}_1 \parallel \mathcal{L}_2 \parallel \mathcal{L}_3\), along with some transversals. The numbers 3, 9, 6, and 8 are present, and two variables \(a\) and \(b\) are marked. The goal is to calculate \(a + b\).

This is a typical problem involving similar triangles formed by parallel lines and transversals. The key to solving this is using the property of similar triangles, which tells us that corresponding segments between the transversals are proportional because the lines are parallel.

Given that \(\mathcal{L}_1 \parallel \mathcal{L}_2 \parallel \mathcal{L}_3\), we can set up a proportion based on the known lengths:

- On the first transversal, the segments are 3, 9, and 6.
- On the second transversal, the corresponding segments are 8, \(a\), and \(b\).

We can set up two proportions:

\[
\frac{3}{8} = \frac{9}{a} = \frac{6}{b}
\]

Now, solving for \(a\) and \(b\):

1. For \(a\):
\[
\frac{3}{8} = \frac{9}{a} \implies a = \frac{9 \times 8}{3} = 24
\]

2. For \(b\):
\[
\frac{3}{8} = \frac{6}{b} \implies b = \frac{6 \times 8}{3} = 16
\]

Thus, \(a = 24\) and \(b = 16\), so:

\[
a + b = 24 + 16 = 40
\]

### Final Answer:
\(a + b = 40\).

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Do you want further details or have any other questions?

Here are five related questions you could explore:
1. How would the proportions change if the transversals were not parallel?
2. What happens to the ratio if the segments between the lines are not constant?
3. How can we solve a similar problem if the angles between the lines are given instead of lengths?
4. How does the use of proportionality relate to the concept of similar triangles?
5. Can we apply this method to non-linear shapes like circles or parabolas?

**Tip**: Always remember that parallel lines and transversals create similar triangles, which leads to proportional segments.

Calculate a + b given that L1, L2, and L3 are parallel lines with transversals dividing the segments as shown in the image, where certain values are provided.

This problem involves calculating the values of a and b by using proportions formed by parallel lines (L1, L2, L3) and transversals. Learn how to set up and solve the proportionality equations to find a + b. This exercise covers key geometric principles such as similar triangles and the property of proportionality. Suitable for students in Grades 9-11.

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