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Figure below shows ΔADE with a line BC forming another triangle. If line BC is parallel to line DE and the sides of the triangle are as shown, show that

a.

Triangles ABC and ADE are similar

b.

Calculate AD and DE

Let's tackle the problem step by step. Given the triangle \(\Delta ADE\) with a line \(BC\) parallel to \(DE\), forming another triangle \(\Delta ABC\), we need to:

a. Show that \(\Delta ABC\) and \(\Delta ADE\) are similar.
b. Calculate the lengths of \(AD\) and \(DE\).

### Part (a): Showing Similarity

To show that \(\Delta ABC\) and \(\Delta ADE\) are similar, we can use the concept of similar triangles. Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.

1. **Angles:**
   - Since \(BC \parallel DE\), by the Alternate Interior Angles Theorem, \(\angle BAC = \angle DAE\).
   - Also, \(\angle ABC = \angle ADE\), because they are corresponding angles.

2. **Proportional Sides:**
   - Since the triangles share the same angle \(\angle A\) and the two pairs of corresponding angles are equal, the triangles are similar by the AA (Angle-Angle) similarity criterion.

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

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

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

Since \(\Delta ABC \sim \Delta ADE\), the corresponding sides are proportional. 

From similarity,
\[
\frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE}
\]

#### Finding \(AD\):

We know that \(AB\) and \(AD\) are corresponding sides in similar triangles:
\[
\frac{AB}{AD} = \frac{AC}{AE}
\]
Since \(AB = 5 \text{ cm}\) and \(AC = 3 \text{ cm}\), let's denote \(AD\) as \(x\) and \(AE = AC + CE = 3 \text{ cm} + 2.4 \text{ cm} = 5.4 \text{ cm}\):

\[
\frac{5}{x} = \frac{3}{5.4}
\]

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 \(AC = 3 \text{ cm}\), and \(AE = 5.4 \text{ cm}\), let's denote \(DE\) as \(y\):
\[
\frac{4}{y} = \frac{3}{5.4}
\]

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. What are other criteria for proving the similarity of triangles?
2. How can you use proportionality to solve for unknown sides in similar triangles?
3. What is the significance of the AA similarity criterion?
4. How do parallel lines aid in determining triangle similarity?
5. Can you explain the importance of corresponding angles in proving similarity?
6. How would you solve for the sides of a triangle if one of the triangles is an isosceles triangle?
7. What are some real-life applications of similar triangles?
8. How does the Pythagorean theorem relate to similar triangles?

**Tip:** When solving geometry problems, drawing additional lines or marking equal angles can often make relationships clearer and simplify the problem.

Learn how to prove similarity between triangles ABC and ADE using the Angle-Angle (AA) similarity criterion. Calculate side lengths AD and DE based on given triangle dimensions. Suitable for high school students studying geometry.

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