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### 1. **Similarity of Triangles (Problem 1):**
The question asks to prove that the triangles in *Figura 4.120* are similar and to indicate the criteria used. 
- The two triangles share the same angles:
  - \( \angle B = \angle E = 104^\circ \)
  - \( \angle A = \angle D = 47^\circ \)
  - \( \angle C = \angle F = 29^\circ \)
  
Since all corresponding angles are equal, the triangles are similar by the **Angle-Angle (AA) Similarity Criterion**.

### 2. **Determine Sides BC and AC (Problem 2):**
In *Figura 4.121*, two triangles are given with angles and some side lengths.
- Since the triangles are similar (again by AA similarity), the corresponding sides are proportional. To find \( BC \) and \( AC \), we set up proportions based on corresponding sides:
  - \( \frac{BC}{DF} = \frac{AB}{DE} = \frac{AC}{EF} \)
  - Using known side lengths \( AB = 6 \text{ cm}, DE = 36 \text{ cm}, AC = 24 \text{ cm}, EF = 18 \text{ cm} \).

### 3. **Determine Sides of the Second Triangle (Problem 3):**
A triangle has sides 24 cm, 40 cm, and 28 cm, and is similar to another triangle with a perimeter of 46 cm. We need to find the side lengths of the second triangle.
- First, find the perimeter of the first triangle:
  \[
  P_1 = 24 + 40 + 28 = 92 \text{ cm}
  \]
- The scale factor is:
  \[
  \text{Scale factor} = \frac{P_2}{P_1} = \frac{46}{92} = \frac{1}{2}
  \]
- Multiply each side of the first triangle by \( \frac{1}{2} \):
  \[
  24 \times \frac{1}{2} = 12 \text{ cm}, \quad 40 \times \frac{1}{2} = 20 \text{ cm}, \quad 28 \times \frac{1}{2} = 14 \text{ cm}
  \]
- So, the sides of the second triangle are 12 cm, 20 cm, and 14 cm.

### 4. **Calculate BC (Problem 4):**
In *Figura 4.122*, \( E \) is the midpoint of \( AB \) and \( EF \parallel AC \). Using the midpoint theorem, we know that \( EF \) is half the length of \( AC \). Given \( AC = 42 \text{ cm} \), we calculate:
\[
EF = \frac{1}{2} \times 42 = 21 \text{ cm}
\]

Since \( E \) is the midpoint of \( AB \), the line \( EF \) divides \( ABC \) into two similar triangles.

### 5. **Determine Similarity of Triangles (Problem 5):**
Two cases are presented:
- **(a)** By Angle-Angle Similarity, if all three corresponding angles match, the triangles are similar.
- **(b)** For right triangles, if the angles and corresponding side lengths (proportional sides) match, the triangles are similar.

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Here are five follow-up questions based on the exercise:
1. What is the Angle-Angle (AA) similarity criterion?
2. How do you calculate the scale factor between two similar triangles?
3. What is the significance of the midpoint theorem in geometry?
4. How can you use proportional sides to find unknown lengths in similar triangles?
5. Why does having corresponding equal angles guarantee similarity in triangles?

**Tip:** Always check for corresponding angles when determining if two triangles are similar. This is one of the easiest ways to verify similarity.

Various problems related to triangle similarity, proportional sides, and theorems (AA similarity criterion, midpoint theorem) are given in the image.

Solve various geometry problems related to triangle similarity, using the Angle-Angle (AA) Similarity criterion and the Midpoint Theorem. Learn to calculate unknown side lengths of similar triangles through proportionality and apply the midpoint theorem.

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