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Sagot :
Certainly! Let's complete the proof step-by-step.
[tex]\[ \begin{tabular}{|l|l|} \hline \textbf{Statements} & \textbf{Reasons} \\ \hline $AB=3, BC=5$, and $CA=3.5$ & Given \\ $LM=6.3, MN=9$, and $NL=5.4$ & Given \\ \hline $\frac{NL}{AB}=\frac{5.4}{3}$ & Definition of ratio \\ $\frac{MN}{BC}=\frac{9}{5}$ & Substitution property of equality \\ $\frac{LM}{CA}=\frac{6.3}{3.5}$ & Definition of ratio \\ \hline $\frac{5.4}{3}=1.8$ & Division of values \\ $\frac{9}{5}=1.8$ & Division of values \\ $\frac{6.3}{3.5}=1.8$ & Division of values \\ \hline $\frac{NL}{AB}=\frac{MN}{BC}=\frac{LM}{CA}$ & Transitive property of equality (if all ratios are equal) \\ \hline $\triangle ABC \sim \triangle LNM$ & If corresponding side lengths of two triangles are in proportion, then the triangles are similar (by the SSS similarity criterion) \\ \hline \end{tabular} \][/tex]
Thus, we have demonstrated that triangles [tex]\( \triangle ABC \)[/tex] and [tex]\( \triangle LNM \)[/tex] are similar because the ratio of their corresponding sides is equal.
[tex]\[ \begin{tabular}{|l|l|} \hline \textbf{Statements} & \textbf{Reasons} \\ \hline $AB=3, BC=5$, and $CA=3.5$ & Given \\ $LM=6.3, MN=9$, and $NL=5.4$ & Given \\ \hline $\frac{NL}{AB}=\frac{5.4}{3}$ & Definition of ratio \\ $\frac{MN}{BC}=\frac{9}{5}$ & Substitution property of equality \\ $\frac{LM}{CA}=\frac{6.3}{3.5}$ & Definition of ratio \\ \hline $\frac{5.4}{3}=1.8$ & Division of values \\ $\frac{9}{5}=1.8$ & Division of values \\ $\frac{6.3}{3.5}=1.8$ & Division of values \\ \hline $\frac{NL}{AB}=\frac{MN}{BC}=\frac{LM}{CA}$ & Transitive property of equality (if all ratios are equal) \\ \hline $\triangle ABC \sim \triangle LNM$ & If corresponding side lengths of two triangles are in proportion, then the triangles are similar (by the SSS similarity criterion) \\ \hline \end{tabular} \][/tex]
Thus, we have demonstrated that triangles [tex]\( \triangle ABC \)[/tex] and [tex]\( \triangle LNM \)[/tex] are similar because the ratio of their corresponding sides is equal.
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