Triangle L M Q is cut by perpendicular bisector L N. Angle N L Q is 32 degrees and angle L M N is 58 degrees.

To determine whether triangles MNL and QNL are congruent, we need to analyze the information given:

1. Angle NLM = 58 degrees (given).
2. Angle NLQ = 32 degrees (given).
3. LN is the perpendicular bisector of segment LQ, which means it forms right angles with QL and LM.

Since LN is a perpendicular bisector:

• Triangle MNL has:

  • Angle NLM = 58 degrees
  • Angle NLQ = 32 degrees
  • Angle NML = 90 degrees (right triangle)

• Triangle QNL has:

  • Angle NQ = 90 degrees (right triangle)
  • Angle NLQ = 32 degrees
  • Angle NQL (which is congruent to angle NLM) = 58 degrees

Now we can evaluate potential congruence criteria:

1. ASA (Angle-Side-Angle): We have angle NML (90 degrees), angle NLM (58 degrees), and angle NLQ (32 degrees) in triangle MNL. For triangle QNL, we have angles NQL (58 degrees) and NLQ (32 degrees), along with right angle NQ (90 degrees). So we have two angles and the included side (LN) in both triangles.

2. AAS (Angle-Angle-Side): Similarly, both triangles share side LN, and we have two pairs of angles that are equal:

Thus, we can conclude that:

• Yes, Triangle MNL is congruent to Triangle QNL by either ASA or AAS.

So, the correct answer is: Yes, they are congruent by either ASA or AAS.