\[ \begin{array}{l} \mathrm{AB}-4,9 \mathrm{~cm}, \mathrm{AD}=8,5 \mathrm{~cm}, \mathrm{AF}=11,8 \mathrm{~cm} \text { and } \mathrm{AH}=13,9 \mathrm{~cm} \\ \mathrm{AF}=5,4 \mathrm{~cm}, \mathrm{AE}=9,4 \mathrm{~cm}, \mathrm{AG}=13 \mathrm{~cm} \text { and } \mathrm{AJ}=15,3 \mathrm{~cm} \\ \mathrm{CH}=2,1 \mathrm{~cm}, \mathrm{ED}=4 \mathrm{~cm}, \mathrm{GF}=5,5 \mathrm{~cm} \text { and } \mathrm{JH}=6,4 \mathrm{~cm} \end{array} \] 11 Complete the similar triangles by writing the letters of the triangles in the correct order. ADC \( { }^{\text {il }} \) A \( \qquad \) \( A D E \) \( \qquad \) III \( \triangle \) AH; 1.2. Why are fly for triangles similar to esch other? \( \qquad \) sit ing portant to write the leters in the correct order when it comes to similarity (2) 14. Why is it ingertant to write the letters in the correct order when it comes to sinuianity \( \qquad \) \( \qquad \)

To address your questions about similar triangles, let's break them down step by step. ## 1. Completing the Similar Triangles by Writing the Letters in Correct Order When dealing with similar triangles, it's essential to match corresponding vertices in the **same order**. This ensures that each angle and side corresponds correctly between the two triangles. ### Example: Suppose you have two triangles, **ΔADC** and **ΔAHE**. To express that these triangles are similar, you must order the vertices so that: - **A** corresponds to **A** - **D** corresponds to **H** - **C** corresponds to **E** Thus, the correct way to denote their similarity would be: \[ \triangle ADC \sim \triangle AHE \] This notation ensures that: - **Angle A** in ΔADC corresponds to **Angle A** in ΔAHE. - **Angle D** in ΔADC corresponds to **Angle H** in ΔAHE. - **Angle C** in ΔADC corresponds to **Angle E** in ΔAHE. ### Steps to Complete Similar Triangles: 1. **Identify Corresponding Angles:** Ensure that each angle in one triangle corresponds to an angle in the other triangle. 2. **Match the Order of Vertices:** Start with one vertex and follow the correspondence for the remaining vertices in sequence. 3. **Use the Correct Notation:** Use the similarity symbol (∼) and list the vertices in the order that reflects their correspondence. ## 2. Why Are These Triangles Similar to Each Other? Triangles are similar if all their corresponding angles are equal and the lengths of their corresponding sides are in proportion. There are several criteria to establish similarity: 1. **AA (Angle-Angle) Criterion:** If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. 2. **SSS (Side-Side-Side) Criterion:** If the corresponding sides of two triangles are in proportion, the triangles are similar. 3. **SAS (Side-Angle-Side) Criterion:** If an angle of one triangle is equal to an angle of another triangle, and the sides including these angles are in proportion, the triangles are similar. In your case, based on the provided side lengths, you can verify the similarity by checking if the sides are proportional. For instance: \[ \frac{\mathrm{AD}}{\mathrm{AH}} = \frac{8.5}{13.9} \approx 0.612 \\ \frac{\mathrm{CD}}{\mathrm{HE}} = \frac{?}{?} \quad (\text{You would need to provide these lengths}) \] If the ratios of all corresponding sides are equal and all corresponding angles are equal, the triangles are similar. ## 3. Importance of Writing the Letters in the Correct Order When Indicating Similarity The **order of the vertices** in the triangle's notation is crucial because it defines which vertices correspond to each other between the two triangles. Incorrect ordering can lead to mismatched correspondences, making the similarity statement invalid. ### Why Correct Order Matters: - **Correspondence of Angles:** Each vertex represents an angle. Incorrect ordering can imply that non-corresponding angles are equal. - **Proportional Sides:** The sides between corresponding vertices must be proportional. Mismatched ordering can disrupt this proportionality. - **Clarity:** Maintaining a consistent order ensures that anyone reading the notation understands the intended correspondence without confusion. ### Example of Incorrect vs. Correct Order: - **Incorrect:** \(\triangle ABC \sim \triangle ACB\) - Here, **B** corresponds to **C** and **C** corresponds to **B**, which may not be true. - **Correct:** \(\triangle ABC \sim \triangle AHE\) - Ensures that **A** corresponds to **A**, **B** to **H**, and **C** to **E**, maintaining proper correspondence. ## Summary - **Order Matters:** Always list the vertices in the same order to maintain correct correspondence. - **Verify Criteria:** Use established similarity criteria (AA, SSS, SAS) to confirm similarity. - **Consistent Notation:** Proper notation communicates your reasoning clearly and accurately. By following these guidelines, you can accurately complete and denote similar triangles in your geometry work.