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In the diagram, side AB ∥ DE, AB = 20 m, DE = 34 m and BC = 9 m. Find the length of CD. Hint: The two triangles are similar. Why? Which sides are corresponding?

Thank you for the diagram! Let's work through the solution step-by-step. 

### Step 1: Recognizing Similar Triangles  
- Since **AB ∥ DE** and both lines are intersected by transversal lines \( AC \) and \( BE \), corresponding angles are equal by the **alternate interior angles theorem**.  
- Therefore, \( \triangle ABC \) is similar to \( \triangle CDE \) by the **AA similarity criterion** (two corresponding angles are equal).

---

### Step 2: Setting Up the Corresponding Ratios  
Since the triangles are similar, the ratios of the corresponding sides will be equal. The corresponding sides are:

1. \( AB \) (in \( \triangle ABC \)) corresponds to \( DE \) (in \( \triangle CDE \))
2. \( BC \) (in \( \triangle ABC \)) corresponds to \( CD \) (in \( \triangle CDE \))

Thus, we can set up the following proportion:

\[
\frac{AB}{DE} = \frac{BC}{CD}
\]

---

### Step 3: Substituting the Known Values  
Given:
- \( AB = 20 \, m \)
- \( DE = 34 \, m \)
- \( BC = 9 \, m \)

Substitute these into the proportion:

\[
\frac{20}{34} = \frac{9}{CD}
\]

---

### Step 4: Solving for \( CD \)  
Cross-multiplying:

\[
20 \cdot CD = 34 \cdot 9
\]

\[
20 \cdot CD = 306
\]

Now, divide both sides by 20:

\[
CD = \frac{306}{20} = 15.3 \, m
\]

---

### Final Answer  
The length of \( CD \) is:

\[
CD = 15.3 \, m
\]

---

Would you like further clarification or additional details on how we applied similarity here?

---

### Five Related Questions:
1. How do we prove two triangles are similar using the AA similarity criterion?
2. If one of the angles in \( \triangle ABC \) is 50°, what would be the corresponding angle in \( \triangle CDE \)?
3. What would happen if \( AB \) and \( DE \) were not parallel? Would the triangles still be similar?
4. Can the ratio of areas of similar triangles be determined from the sides? If yes, how?
5. How do we determine if two lines are parallel using angles created by a transversal?

### Tip:
When working with similar triangles, always align the corresponding sides carefully. This ensures your proportions are accurate!

In the diagram, side AB ∥ DE, AB = 20 m, DE = 34 m, and BC = 9 m. Find the length of CD. Hint: The two triangles are similar. Why? Which sides are corresponding?

This problem involves triangle similarity with AB ∥ DE and AB = 20 m, DE = 34 m, BC = 9 m. Using proportions and the AA similarity criterion, learn how to solve for the length of CD step by step. Suitable for students in Grades 8-10.

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