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This problem involves solving for the missing variable(s) in a triangle setup with parallel lines and proportional segments. Here’s the step-by-step approach:

### Problem Analysis:
1. **Given**:
   - Segment \( \overline{CB} = 70 \)
   - Segment \( \overline{KL} = 15 \)
   - \( KD = 9 \), \( DL = 12 \), and the expression for \( CD \) is \( 11x - 2 \).
   - \( BD = 56 \).

2. The segments form similar triangles by the property of parallel lines (\( \overline{KL} \parallel \overline{CB} \)).

### Step-by-Step Solution:

#### Step 1: Use the property of similar triangles
Since \( \triangle KDL \sim \triangle CDB \), the ratios of corresponding sides are equal:
\[
\frac{\overline{KL}}{\overline{CB}} = \frac{\overline{KD}}{\overline{CD}} = \frac{\overline{DL}}{\overline{DB}}.
\]

#### Step 2: Solve for \( CD \) using the ratio \( \frac{KD}{CD} \)
Using the first ratio:
\[
\frac{\overline{KL}}{\overline{CB}} = \frac{\overline{KD}}{\overline{CD}}.
\]
Substitute the known values:
\[
\frac{15}{70} = \frac{9}{11x - 2}.
\]
Simplify the left-hand side:
\[
\frac{3}{14} = \frac{9}{11x - 2}.
\]
Cross-multiply:
\[
3(11x - 2) = 9 \times 14.
\]
Simplify:
\[
33x - 6 = 126.
\]
Solve for \( x \):
\[
33x = 132 \quad \implies \quad x = 4.
\]

#### Step 3: Verify and calculate \( CD \)
Substitute \( x = 4 \) into \( CD = 11x - 2 \):
\[
CD = 11(4) - 2 = 44 - 2 = 42.
\]

### Final Answer:
The missing variable \( x \) is \( \boxed{4} \), and the segment \( CD \) is \( \boxed{42} \).

Let me know if you'd like a detailed breakdown of any specific step!  

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Here are five related questions for further practice:
1. Can you calculate the ratio of the areas of triangles \( \triangle KDL \) and \( \triangle CDB \)?
2. What are the properties of parallel lines and similar triangles?
3. How can the triangle similarity theorem be applied in real-world geometry problems?
4. If \( x = 5 \), what would \( CD \) be in this scenario?
5. What is the sum of all the interior angles in \( \triangle CDB \)?

### Tip:
When working with similar triangles, always ensure the corresponding sides are matched correctly to avoid errors in setting up proportions.

For #9-14, solve for the missing variable(s): Using the diagram where triangles KDL and CDB are similar, solve for x and find the length of segment CD.

Step-by-step solution for solving the missing variable x and calculating CD in a problem involving similar triangles KDL and CDB. Includes triangle similarity and proportional relationships.

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