Question 1 [6] Given $$ \begin{array}{l} \mathrm{AB}=4,9 \mathrm{~cm}, \mathrm{AD}=8.5 \mathrm{~cm}, \mathrm{AF}=11,8 \mathrm{~cm} \text { and } \mathrm{AH}=13,9 \mathrm{~cm} \ \mathrm{AC}=5,4 \mathrm{~cm} . \mathrm{AE}=9,4 \mathrm{~cm} . \mathrm{AG}=15 \mathrm{~cm} \text { and } \mathrm{AJ}=15,3 \mathrm{~cm} \ \mathrm{CB}=23 \mathrm{~cm}, \mathrm{ED}=4 \mathrm{~cm} . \mathrm{GF}=5,5 \mathrm{~cm} \text { and } \mathrm{II}=6,4 \mathrm{~cm} \end{array} $$ 11. Coryplece the similar triangles by writing the lecters of the trangles in the correct order. $$ \triangle A B C\|\| $$ $$ \qquad $$ III $$ \Delta $$ $$ \qquad $$ ||| $$ \Delta $$ $$ \qquad $$ (3) 12 Why are the four manglo similar to each other?

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UpStudy AI · Accepted Answer

Sure, let's break down and address each part of your question regarding similar triangles.

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### **Question 11 [6]**
**Complete the similar triangles by writing the letters of the triangles in the correct order.**

Given triangles with various side lengths, we need to identify which triangles are similar by ensuring that their corresponding angles and sides are in the same order.

**Steps to Determine Similar Triangles:**

1. **Identify Corresponding Sides:**
   - Similar triangles have sides that are proportional. Compare the given side lengths to find proportional relationships.
   
2. **Match Corresponding Angles:**
   - The order of the letters in the triangle names should correspond to their angles. For example, if angle **A** in one triangle corresponds to angle **D** in another, then the first vertex of each triangle name should match accordingly.

3. **Write the Similarity Statement:**
   - Use the tilde symbol (`~`) to denote similarity.
   - Ensure the order of the letters reflects the correspondence of the angles and sides.

**Example Completion:**
Suppose you have four triangles: ΔABC, ΔDEF, ΔGHI, and ΔJKL. If after analysis, you determine that:

- ΔABC is similar to ΔDEF
- ΔDEF is similar to ΔGHI
- ΔGHI is similar to ΔJKL

Then, the similarity can be written as:
$$
\triangle ABC \sim \triangle DEF \sim \triangle GHI \sim \triangle JKL
$$

**Note:** Without a specific diagram or additional information about the relationships between the triangles, this example serves as a general guide. Ensure that you match the vertices in the order that maintains the correspondence of angles and proportional sides.

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### **Question 12**
**Why are the four triangles similar to each other?**

**Reasons for Similarity:**

1. **Equal Corresponding Angles:**
   - If all corresponding angles of the four triangles are equal, the triangles are similar by the **Angle-Angle (AA) Similarity Postulate**.

2. **Proportional Corresponding Sides:**
   - If the lengths of corresponding sides are in the same proportion, the triangles are similar by the **Side-Side-Side (SSS) Similarity Criterion** or the **Side-Angle-Side (SAS) Similarity Criterion** if one pair of corresponding angles is equal.

3. **Consistent Ratios:**
   - The ratios of the lengths of corresponding sides are consistent across all triangles. For instance, if in ΔABC and ΔDEF, $$ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} $$, and this proportionality holds when comparing to the other triangles, similarity is established.

4. **Geometric Configuration:**
   - Sometimes, the geometric arrangement (like parallel lines cut by transversals) ensures that corresponding angles are equal, leading to similarity.

**Conclusion:**
The four triangles are similar to each other because they each have their corresponding angles equal and their corresponding sides are proportional. This fulfills the necessary conditions (AA, SSS, or SAS) for triangle similarity.

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**Summary:**
- **For Question 11:** Ensure that when listing similar triangles, the order of the vertices corresponds to equal angles and proportional sides.
- **For Question 12:** The similarity arises from equal corresponding angles and proportional corresponding sides among all four triangles.

If you have a specific diagram or additional details, feel free to share them for a more tailored explanation!
To determine if four triangles are similar, check if their corresponding angles are equal and their corresponding sides are proportional. If both conditions are met, the triangles are similar.

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