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The given image shows a quadrilateral \( ABCD \) where \( AB = 18 \, \text{cm} \), \( AC = 12 \, \text{cm} \), \( AD = 16 \, \text{cm} \), and \( CD = 20 \, \text{cm} \). We are asked to find the lengths of segments \( BE \) and \( DE \) where \( E \) lies on \( AD \) and \( B \) lies on \( AC \).

Since \( BE \) and \( DE \) are inside the triangle, and \( E \) is on the diagonal \( AD \), we can use the concept of similar triangles to find their lengths. Given that \( BE \parallel CD \), triangle \( ABE \) is similar to triangle \( ACD \).

### Using Similarity of Triangles:
1. **\( \triangle ABE \sim \triangle ACD \)**:
   - Since \( \triangle ABE \sim \triangle ACD \), the ratio of corresponding sides is equal:
   \[
   \frac{AB}{AC} = \frac{BE}{CD} = \frac{AE}{AD}
   \]
   
   Let's denote \( BE = x \) and \( DE = y \).

   \[
   \frac{18}{12} = \frac{x}{20}
   \]
   Simplifying the left-hand side:
   \[
   \frac{3}{2} = \frac{x}{20}
   \]
   Solving for \( x \):
   \[
   x = \frac{3}{2} \times 20 = 30 \, \text{cm}
   \]

2. **Finding \( DE \)**:
   Since \( BE \) and \( DE \) form a line parallel to the base \( CD \) in \( \triangle ABC \):
   \[
   \frac{BE}{CD} = \frac{AE}{AD}
   \]
   Again using the triangle similarity, let's denote \( AE = z \):
   \[
   \frac{18}{12} = \frac{z}{16}
   \]
   Solving for \( z \):
   \[
   z = \frac{18 \times 16}{12} = 24 \, \text{cm}
   \]
   
   \( DE = z - 16 = 24 - 16 = 8 \, \text{cm} \).

So the lengths are:
- \( BE = 30 \, \text{cm} \)
- \( DE = 8 \, \text{cm} \)

Do you have any questions or would you like further details?

Here are five related questions to explore this topic further:
1. How do we prove that triangles are similar?
2. Can the Pythagorean theorem be applied in this problem to find any other lengths?
3. What would happen if \( BE \) and \( DE \) were not parallel to \( CD \)?
4. How do you calculate the area of \( \triangle ABC \) using the given lengths?
5. What are the properties of quadrilaterals, and how do they apply to this problem?

**Tip:** When working with similar triangles, always check the ratios of corresponding sides to ensure the calculations are accurate.

Explore how to find the lengths of segments BE and DE in quadrilateral ABCD using the concept of similar triangles. Detailed step-by-step solution provided with calculations and explanations. Suitable for high school students studying geometry.

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