Use geometric techniques to solve  problems

Sure, let's approach the problem using geometric techniques.

Given the triangle \( \Delta ADE \) with line segment \( BC \) parallel to \( DE \) forming \( \Delta ABC \), let's solve it using similar triangles and proportions.

### Part (a): Showing Similarity

We need to show that \( \Delta ABC \) and \( \Delta ADE \) are similar.

1. **Parallel Lines:**
   Since \( BC \parallel DE \), the corresponding angles are equal:
   - \( \angle BAC = \angle DAE \) (corresponding angles)
   - \( \angle ABC = \angle ADE \) (corresponding angles)

2. **Common Angle:**
   - \( \angle A \) is common to both triangles.

Since two angles of one triangle are equal to two angles of another triangle, the triangles are similar by the Angle-Angle (AA) similarity criterion.

Thus, \( \Delta ABC \sim \Delta ADE \).

### Part (b): Calculating \( AD \) and \( DE \)

Given:
- \( AB = 5 \text{ cm} \)
- \( BC = 4 \text{ cm} \)
- \( AC = 3 \text{ cm} \)
- \( CE = 2.4 \text{ cm} \)

#### Finding \( AD \):

From the similarity of triangles \( \Delta ABC \sim \Delta ADE \), the sides are proportional:
\[
\frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE}
\]

Let's denote \( AD \) as \( x \) and \( DE \) as \( y \).

Since \( AE = AC + CE \):
\[
AE = 3 \text{ cm} + 2.4 \text{ cm} = 5.4 \text{ cm}
\]

Using the sides \( AB \) and \( AD \):
\[
\frac{AB}{AD} = \frac{AC}{AE}
\]
\[
\frac{5}{x} = \frac{3}{5.4}
\]

Cross-multiplying to solve for \( x \):
\[
5 \times 5.4 = 3x
\]
\[
27 = 3x
\]
\[
x = 9 \text{ cm}
\]

Therefore, \( AD = 9 \text{ cm} \).

#### Finding \( DE \):

Using the proportionality of sides:
\[
\frac{BC}{DE} = \frac{AC}{AE}
\]

Given \( BC = 4 \text{ cm} \) and \( AC = 3 \text{ cm} \):
\[
\frac{4}{y} = \frac{3}{5.4}
\]

Cross-multiplying to solve for \( y \):
\[
4 \times 5.4 = 3y
\]
\[
21.6 = 3y
\]
\[
y = 7.2 \text{ cm}
\]

Therefore, \( DE = 7.2 \text{ cm} \).

### Summary

- \( \Delta ABC \) and \( \Delta ADE \) are similar by AA similarity.
- \( AD = 9 \text{ cm} \)
- \( DE = 7.2 \text{ cm} \)

Would you like any more details or have any other questions?

Here are 8 related questions to further your understanding:
1. How do you identify similar triangles in complex geometric figures?
2. What are the properties of similar triangles that can be useful in problem-solving?
3. How can the concept of parallel lines be used to find unknown angles in geometry?
4. What are the other methods to prove triangles are similar besides AA similarity?
5. How do you use the properties of similar triangles in coordinate geometry?
6. Can you explain how to solve problems involving similar triangles using the ratio of areas?
7. How does the concept of similarity help in real-world applications like map reading and architecture?
8. What is the role of corresponding angles in determining the similarity of triangles?

**Tip:** Practice drawing accurate diagrams and marking known lengths and angles. This helps in visualizing the problem better and makes it easier to apply geometric principles.

Explore how geometric techniques, specifically involving similar triangles and proportions, can be used to solve problems. This detailed solution demonstrates the application of the Angle-Angle (AA) similarity criterion in determining unknown sides of triangles. Suitable for high school students studying geometry.

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