Question 1 [6] Given : \[ \begin{array}{l}\mathrm{AB}=4,9 \mathrm{~cm}, \mathrm{AD}=8,5 \mathrm{~cm}, \mathrm{AF}-11,8 \mathrm{~cm} \text { and } \mathrm{AH}=13,9 \mathrm{~cm} \\ \mathrm{AC}=5,4 \mathrm{~cm}, \mathrm{AE}=9,4 \mathrm{~cm}, \mathrm{AG}=13 \mathrm{~cm} \text { and } \mathrm{AJ}=15,3 \mathrm{~cm} \\ \mathrm{CB}=2,3 \mathrm{~cm}, \mathrm{ED}=4 \mathrm{~cm}, \mathrm{GF}=5,5 \mathrm{~cm} \text { and } \mathrm{JH}=6,4 \mathrm{~cm}\end{array} \] 1.1 Complete the similar triangles by writing the letters of the triangles in the correct order. 1.2 Why are the four triangles similar to each other? 1.2 Why is it important to write the letters in the correct order when it comes to similarity

1.1 The triangles are formed by vertex \(A\) and two other points on each line. Using the given side lengths we can identify the four triangles as follows: - In the first triangle, the sides are \(AB=4.9\,\mathrm{cm}\), \(AC=5.4\,\mathrm{cm}\), and \(CB=2.3\,\mathrm{cm}\). This triangle is written as \(\triangle ABC\). - In the second triangle, the sides are \(AD=8.5\,\mathrm{cm}\), \(AE=9.4\,\mathrm{cm}\), and \(ED=4\,\mathrm{cm}\). This triangle is written as \(\triangle ADE\). - In the third triangle, the sides are \(AF=11.8\,\mathrm{cm}\), \(AG=13\,\mathrm{cm}\), and \(GF=5.5\,\mathrm{cm}\). This triangle is written as \(\triangle AFG\). - In the fourth triangle, the sides are \(AH=13.9\,\mathrm{cm}\), \(AJ=15.3\,\mathrm{cm}\), and \(JH=6.4\,\mathrm{cm}\). This triangle is written as \(\triangle AHJ\). 1.2 All four triangles are similar because: - They share a common vertex at \(A\); therefore, they all have the same angle at \(A\). - Their other sides are constructed by points taken along straight lines that are in the same direction (or form parallel lines in the complete figure), which ensures that the angles at the remaining vertices are equal, too. - By the Angle-Angle (AA) criterion, having two corresponding angles equal ensures that the triangles are similar. This similarity guarantees that the ratios of the corresponding sides are constant, for example: \[ \frac{AB}{AC}=\frac{AD}{AE}=\frac{AF}{AG}=\frac{AH}{AJ} \] 1.3 It is essential to write the letters in the correct order when stating that triangles are similar because: - The order of the letters establishes the correspondence between vertices. For example, in \(\triangle ABC\) and \(\triangle ADE\), the first vertex \(A\) corresponds to \(A\), the second vertex \(B\) corresponds to \(D\), and the third vertex \(C\) corresponds to \(E\). - Proper correspondence ensures that the correct sides are compared when setting up the proportional relationships. In our example, it guarantees that we compare \(\frac{AB}{AD}\), \(\frac{AC}{AE}\), and \(\frac{BC}{DE}\) correctly. - If the vertices were not ordered correctly, the established proportionality between corresponding sides might not hold, leading to incorrect conclusions about the similarity of the triangles.