Question 1 [6] Given \[ \begin{array}{l} \mathrm{AB}=4,9 \mathrm{~cm}, \mathrm{AD}=8,5 \mathrm{~cm}, \mathrm{AF}=11,8 \mathrm{~cm} \text { and } \mathrm{AH}=13,9 \mathrm{~cm} \\ \mathrm{AC}=5,4 \mathrm{~cm} \quad \mathrm{AE}=9,4 \mathrm{~cm}, \mathrm{AG}=13 \mathrm{~cm} \text { and } \mathrm{AJ}=15,3 \mathrm{~cm} \\ \mathrm{CB}=2,3 \mathrm{~cm} \quad \mathrm{ED}=4 \mathrm{~cm} \quad \mathrm{GF}=5,5 \mathrm{~cm} \text { and } \quad \mathrm{JH}=6,4 \mathrm{~cm} \end{array} \] 11 Complete the similar triangles by writing the letters of the triangles in the correct onder \[ \triangle A B C\|\| \] \( \qquad \) III \( \triangle \) \( \qquad \) II \( \qquad \) (3) 12 Why are the frour triangles similar to each other? \( \qquad \) 13 Why is it important to write the letters in the carroct order when it comes to simsilarity (2) \( \qquad \) \( \qquad \)

### Question 11 **Complete the similar triangles by writing the letters of the triangles in the correct order:** Given the triangles based on the provided measurements, the similar triangles can be identified as follows: \[ \triangle ABC \sim \triangle ADE \sim \triangle AFG \sim \triangle AJH \] **Explanation:** - **\(\triangle ABC\):** Base triangle with sides AB, BC, and AC. - **\(\triangle ADE\):** Shares vertex A with \(\triangle ABC\) and has sides AD, DE, and AE. - **\(\triangle AFG\):** Shares vertex A with the previous triangles and has sides AF, FG, and AG. - **\(\triangle AJH\):** Shares vertex A and has sides AJ, JH, and AH. ### Question 12 **Why are the four triangles similar to each other?** The four triangles \(\triangle ABC\), \(\triangle ADE\), \(\triangle AFG\), and \(\triangle AJH\) are similar to each other because: 1. **Corresponding Angles are Equal:** Each pair of triangles shares the same angles at vertex A. Additionally, the angles formed by the intersecting lines ensure that all corresponding angles in each triangle are equal. 2. **Proportional Corresponding Sides:** The lengths of corresponding sides of the triangles are in the same proportion. For example: - \( \frac{AB}{AD} = \frac{4.9}{8.5} \approx 0.576 \) - \( \frac{AD}{AF} = \frac{8.5}{11.8} \approx 0.720 \) - \( \frac{AF}{AH} = \frac{11.8}{13.9} \approx 0.848 \) While the exact ratios may vary slightly due to measurement precision, the consistent proportionality across all corresponding sides confirms similarity. **Therefore, by the Angle-Angle (AA) similarity criterion and the proportionality of corresponding sides, all four triangles are similar to each other.** ### Question 13 **Why is it important to write the letters in the correct order when it comes to similarity?** Writing the letters of similar triangles in the correct order is crucial because: 1. **Accurate Correspondence of Vertices:** The order of the letters denotes which vertices correspond to each other. For instance, in \(\triangle ABC \sim \triangle ADE\), vertex A corresponds to A, B to D, and C to E. This ensures that angles and sides are correctly matched. 2. **Preservation of Proportional Relationships:** Correct ordering maintains the proportional relationships between corresponding sides. Misordering can lead to incorrect conclusions about similarity, as sides and angles may no longer correspond appropriately. 3. **Clarity and Consistency:** Consistent ordering facilitates clear communication and understanding, especially when multiple similar triangles are involved. It helps avoid confusion and ensures that all geometric properties are accurately conveyed. **In summary, correct letter ordering is essential to properly establish and communicate the similarity between triangles by ensuring that corresponding parts align correctly.**