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Complete the proof.
\begin{tabular}{|l|l|}
\hline Statements & Reasons \\
\hline [tex]$AB = 3$[/tex], [tex]$BC = 5$[/tex], and [tex]$CA = 3.5$[/tex] & Given \\
[tex]$LM = 6.3$[/tex], [tex]$MN = 9$[/tex], and [tex]$NL = 5.4$[/tex] & Given \\
\hline [tex]$\frac{NL}{AB} = \frac{5.4}{3}$[/tex] & Corresponding sides of similar triangles \\
[tex]$\frac{MN}{BC} = \frac{9}{5}$[/tex] & Corresponding sides of similar triangles \\
[tex]$\frac{LM}{CA} = \frac{6.3}{3.5}$[/tex] & Corresponding sides of similar triangles \\
\hline [tex]$\frac{5.4}{3} = 1.8$[/tex] & Simplification \\
[tex]$\frac{9}{5} = 1.8$[/tex] & Simplification \\
[tex]$\frac{6.3}{3.5} = 1.8$[/tex] & Simplification \\
\hline [tex]$\frac{NL}{AB} = \frac{MN}{BC} = \frac{LM}{CA}$[/tex] & All ratios are equal \\
\hline [tex]$\triangle ABC \sim \triangle LNM$[/tex] & By the definition of similar triangles \\
\hline
\end{tabular}

Sagot :

GinnyAnsweridn
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.
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