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Let's solve the given problems step by step:

### Question 4:
In triangles \( \Delta ABC \) and \( \Delta DEF \), it is given that:
- \( \angle F = \angle C \)
- \( \angle B = \angle E \)
- \( AB = \frac{1}{2} DE \)

#### Analyzing:
We are given two pairs of equal angles:
- \( \angle F = \angle C \)
- \( \angle B = \angle E \)

So, by the **AA (Angle-Angle) Similarity Criterion**, the triangles are similar. However, the sides are not proportional for congruence since \( AB = \frac{1}{2} DE \), which means they are **not congruent**.

Thus, the two triangles are **similar but not congruent**.

**Answer**: (b) Similar, but not congruent.

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### Question 5:
In the figure, it is given that \( DE \parallel AC \) and \( DF \parallel AE \), and we need to find the ratio \( \frac{BF}{FE} \).

#### Applying the Basic Proportionality Theorem (Thales Theorem):
Since \( DE \parallel AC \) and \( DF \parallel AE \), the triangles formed are similar. Using the theorem, we know that the ratios of corresponding segments of similar triangles are equal.

From the diagram:
- For \( DE \parallel AC \), the triangles \( \Delta BFE \) and \( \Delta BEC \) are similar.
- Hence, \( \frac{BF}{FE} = \frac{BE}{EC} \).

**Answer**: (B) \( \frac{BE}{EC} \).

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Would you like further details or explanations on any part? 

Here are 5 related questions to expand the information:
1. How do you prove the AA similarity criterion in triangles?
2. What are other similarity theorems for triangles apart from AA?
3. Can similar triangles be congruent under specific conditions?
4. What is the Basic Proportionality Theorem, and how is it applied in different geometrical problems?
5. How do parallel lines in triangles help in solving proportionality problems?

**Tip**: In geometry, identifying similar triangles can simplify complex figures by allowing the use of proportionality rules, which helps in finding unknown side lengths and ratios efficiently.

In triangles ABC and DEF, ∠F = ∠C, ∠B = ∠E, and AB = 1/2 DE. Then the two triangles are: (a) Congruent, but not similar (b) Similar, but not congruent (c) Neither congruent nor similar (d) Congruent as well as similar. In the figure below, DE || AC and DF || AE. Which of these is equal to BF/FE? (A) DF/AE (B) BE/EC (C) BA/AC (D) FE/EC

This guide explains how to determine similarity and proportions in triangles using the AA similarity criterion and the Basic Proportionality Theorem. It covers triangle ABC and DEF, where ∠F = ∠C, ∠B = ∠E, and AB = 1/2 DE, and solves for the ratio BF/FE with parallel lines DE || AC and DF || AE.

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