Geometric diagram of a triangle with side lengths and a variable x on the base.

We are given a triangle with a line drawn inside it, parallel to the base. The sides of the big triangle are labeled: left side is 6 + x (since the bottom part is x and top part is 6), right side is 8 + 5 = 13. The inner line is parallel to the base, so we can use the Triangle Proportionality Theorem (also called Basic Proportionality Theorem or Thales’ Theorem).

This theorem says:
> If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those sides proportionally.

So, in this case, the line cuts the left side into segments of length 6 and x, and the right side into segments of length 8 and 5.

That means:

> 6 / x = 8 / 5

Wait — let’s double-check which parts correspond.

Actually, the segment from the top vertex down to the parallel line on the left is 6, and below that is x. On the right, from top to the parallel line is 8, and below that is 5.

Since the line is parallel, the ratios of the corresponding segments should be equal:

> Top segment / Bottom segment on left = Top segment / Bottom segment on right

So:

> 6 / x = 8 / 5

Now solve for x.

Cross-multiply:

> 6 * 5 = 8 * x
> 30 = 8x

Divide both sides by 8:

> x = 30 / 8
> x = 15 / 4
> x = 3.75

Let me verify: Is 6/3.75 equal to 8/5?

6 ÷ 3.75 = 1.6
8 ÷ 5 = 1.6 → Yes, they match.

Alternatively, we could think of the whole left side as 6 + x, and whole right side as 8 + 5 = 13. Then the ratio of the top part to the whole side should be the same on both sides:

> 6 / (6 + x) = 8 / 13

Let’s try solving this way to check.

Cross-multiply:

> 6 * 13 = 8 * (6 + x)
> 78 = 48 + 8x
> 78 - 48 = 8x
> 30 = 8x
> x = 30/8 = 15/4 = 3.75

Same answer! So it’s correct.

Final Answer:
3.75