Question

are triangles def and lnm similar if ln equals 5, mn equals 3, de equals 10, and fe equals 6? no, the corresponding sides are not proportional no, the corresponding angles are not congruent yes, by the sss similarity postulate

Explanation:

Step1: Check side - length ratios

We need to check if $\frac{LN}{DE}=\frac{MN}{FE}$. Given $LN = 5$, $DE=10$, $MN = 3$, and $FE = 6$. Calculate $\frac{LN}{DE}=\frac{5}{10}=\frac{1}{2}$ and $\frac{MN}{FE}=\frac{3}{6}=\frac{1}{2}$. Also, $\angle F=\angle N = 90^{\circ}$.

Step2: Apply similarity criterion

Since the ratios of the corresponding sides are equal ($\frac{LN}{DE}=\frac{MN}{FE}$) and the included right - angles are congruent, by the SAS (Side - Angle - Side) similarity criterion, the triangles are similar. In this case, also considering the right - angled nature and side - length ratios, we can say by SSS similarity (if we consider the third sides which will also be in proportion) the triangles are similar.

Answer:

Yes, by the SSS Similarity Postulate