Question 1 [6] Given $$ \begin{array}{l} \mathrm{AB}=4,9 \mathrm{~cm}, \mathrm{AD}=8,5 \mathrm{~cm}, \mathrm{AF}=11,8 \mathrm{~cm} \text { and } \mathrm{AH}=13,9 \mathrm{~cm} \ \mathrm{AC}=5,4 \mathrm{~cm}, \mathrm{AE}=9,4 \mathrm{~cm}, \mathrm{AG}=13 \mathrm{~cm} \text { and } \mathrm{AJ}=15,3 \mathrm{~cm} \ \mathrm{CB}=2,3 \mathrm{~cm}, \mathrm{ED}=4 \mathrm{~cm}, \mathrm{GF}=5,5 \mathrm{~cm} \text { and } J H=6.4 \mathrm{~cm} \end{array} $$ 1.1 Complete the similar triangles by writing the letters of the triangles in the correct order. $$ \triangle A B C $$ ||| $$ \triangle $$ $$ \qquad $$ III $$ \triangle $$ $$ \qquad $$ ||| $$ \triangle $$ $$ \qquad $$ .2 Why are the four triangles similar to each other? $$ \qquad $$ 3 Why is it important to write the letters in the correct order when it comes to similarity $$ \qquad $$ $$ \qquad $$

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**Question 1**

Given the lengths of various segments, we can analyze the similarity of the triangles based on proportional side lengths and corresponding angles.

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**1.1 Complete the similar triangles by writing the letters of the triangles in the correct order.**

Assuming the four triangles in question are labeled as $$ \triangle ABC $$, $$ \triangle ADE $$, $$ \triangle AFG $$, and $$ \triangle AHJ $$, the similarity can be expressed as follows:

$$
\triangle ABC \sim \triangle ADE \sim \triangle AFG \sim \triangle AHJ
$$

**Explanation:**
- The similarity symbol $$ \sim $$ indicates that the triangles are similar.
- The order of the letters (vertices) is crucial as it ensures that corresponding angles and sides are matched correctly across all triangles.

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**1.2 Why are the four triangles similar to each other?**

The four triangles are similar to each other because they satisfy the **AA (Angle-Angle) Similarity Criterion**. Specifically:

1. **Equal Corresponding Angles:** Each pair of triangles shares two angles that are equal. Since the sum of the angles in a triangle is always $$180^\circ$$, if two angles are equal, the third must also be equal.

2. **Proportional Corresponding Sides:** The lengths of the corresponding sides are in the same proportion. For example:
   - $$ \frac{AB}{AD} = \frac{4.9\, \text{cm}}{8.5\, \text{cm}} $$
   - $$ \frac{AC}{AE} = \frac{5.4\, \text{cm}}{9.4\, \text{cm}} $$
   
   These ratios are consistent across all corresponding sides, reinforcing the similarity between the triangles.

**Conclusion:**
Because both the corresponding angles are equal and the corresponding sides are proportional, the triangles satisfy the conditions for similarity.

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**1.3 Why is it important to write the letters in the correct order when it comes to similarity?**

Writing the letters of the triangles in the correct order when stating similarity is **crucial** for the following reasons:

1. **Ensures Correct Correspondence:**
   - The order of the vertices determines which angles and sides correspond to each other. For instance, in $$ \triangle ABC \sim \triangle ADE $$, vertex $$ A $$ corresponds to vertex $$ A $$, $$ B $$ to $$ D $$, and $$ C $$ to $$ E $$.
   
2. **Prevents Misinterpretation:**
   - Incorrect ordering can lead to mismatched angles and sides, which invalidates the similarity statement. For example, writing $$ \triangle ABC \sim \triangle AED $$ would incorrectly associate vertex $$ B $$ with $$ E $$ and $$ C $$ with $$ D $$, potentially leading to incorrect conclusions.

3. **Facilitates Accurate Calculations:**
   - When applying similarity for solving problems (like finding unknown side lengths or angles), having the correct correspondence ensures that proportionate relationships are applied accurately.

**Example:**
- Correct Order: $$ \triangle ABC \sim \triangle ADE $$ implies $$ AB $$ corresponds to $$ AD $$, $$ BC $$ to $$ DE $$, and $$ AC $$ to $$ AE $$.
- Incorrect Order: $$ \triangle ABC \sim \triangle AED $$ would mismatch the sides and angles, leading to incorrect ratios and results.

**Conclusion:**
Maintaining the correct order of vertices in similarity statements preserves the integrity of the correspondence between the triangles' parts, ensuring accurate and reliable geometric analyses.

---

**Summary:**
- **1.1:** The similar triangles are $$ \triangle ABC \sim \triangle ADE \sim \triangle AFG \sim \triangle AHJ $$.
- **1.2:** They are similar because they have equal corresponding angles and proportional corresponding sides (AA Criterion).
- **1.3:** Correct ordering of vertices ensures proper correspondence of angles and sides, which is essential for validating similarity and performing accurate calculations.
**Question 1** Given the lengths of various segments, we can analyze the similarity of the triangles based on proportional side lengths and corresponding angles. --- **1.1 Complete the similar triangles by writing the letters of the triangles in the correct order.** $$ \triangle ABC \sim \triangle ADE \sim \triangle AFG \sim \triangle AHJ $$ **1.2 Why are the four triangles similar to each other?** The triangles are similar because they have equal corresponding angles and proportional corresponding sides, satisfying the AA (Angle-Angle) Similarity Criterion. **1.3 Why is it important to write the letters in the correct order when it comes to similarity?** Writing the letters in the correct order ensures that corresponding angles and sides are matched correctly, which is essential for accurate similarity statements and calculations. **Summary:** - The four triangles are similar because they have equal corresponding angles and proportional sides. - Writing the vertices in the correct order is crucial to ensure accurate correspondence between the triangles' parts.

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