Question

are triangles def and lnm similar if ln equals 5, mn equals 3, de equals 10, and fe equals 6? image of two right triangles: def with right angle at e, lnm with right angle at n options: no, the corresponding sides are not proportional; no, the corresponding angles are not congruent; yes, by the sss similarity postulate; yes, by the sas similarity postulate

Explanation:

Step1: Identify right angles

Triangles \( DEF \) and \( LNM \) are right - angled triangles. In \( \triangle DEF \), \( \angle E = 90^{\circ} \) and in \( \triangle LNM \), \( \angle N=90^{\circ} \), so \( \angle E=\angle N \).

Step2: Check side ratios

We have \( LN = 5 \), \( DE=10 \), \( MN = 3 \), \( FE = 6 \).
Calculate the ratio of \( DE\) to \( LN \): \( \frac{DE}{LN}=\frac{10}{5} = 2 \)
Calculate the ratio of \( FE\) to \( MN \): \( \frac{FE}{MN}=\frac{6}{3}=2 \)

Since we have two sides in proportion (\( \frac{DE}{LN}=\frac{FE}{MN} = 2\)) and the included angles (\( \angle E\) and \( \angle N\)) are congruent (both are right angles), by the SAS (Side - Angle - Side) Similarity Postulate, the two triangles are similar.

Answer:

Yes, by the SAS Similarity Postulate