Are triangles DEF and LNM similar if LN equals 4, MN equals 3, DE equals 12, and FE equals 9?

Steps to Solve

1. Identify Corresponding Sides and Angles
   First, we need to identify the respective sides and angles of triangles DEF and LNM. Let's label the sides as:

• $DE$, $EF$, $FD$ for triangle DEF
• $LN$, $NM$, $ML$ for triangle LNM

2. Measure the Sides
   We will need the lengths of the corresponding sides from both triangles. Assume we have:

• $DE = a$, $EF = b$, $FD = c$
• $LN = x$, $NM = y$, $ML = z$

3. Check Proportions for SSS Similarity
   To use the SSS (Side-Side-Side) similarity postulate, calculate the ratios of corresponding sides:
   $$ \frac{DE}{LN} = \frac{EF}{NM} = \frac{FD}{ML} $$
   If all these ratios are equal, then the triangles are similar.

4. Check Angles for SAS Similarity
   Alternatively, to use the SAS (Side-Angle-Side) similarity postulate, we must check one pair of corresponding angles. Assume:

• $\angle D = \angle L$ (if given)
  We would then need to confirm:
  $$ \frac{DE}{LN} = \frac{EF}{NM} $$
  If both conditions are satisfied, then the triangles are similar.

5. Conclusion
   Based on the results from either the SSS or SAS tests, conclude if triangles DEF and LNM are similar.