This problem involves understanding the relationship between the angles and sides of a triangle, specifically using the triangle inequality and the property that larger angles are opposite longer sides.

Given:

1. $\triangle KLM$
2. $m\angle K = 22^\circ$
3. $m\angle L = 38^\circ$

Steps to Solve:

1. Find $m\angle M$:

   • The sum of the angles in a triangle is $180^\circ$. Therefore: $m\angle M = 180^\circ - m\angle K - m\angle L = 180^\circ - 22^\circ - 38^\circ = 120^\circ$

2. Rank the Angles:

   • $m\angle M = 120^\circ$ (largest angle)
   • $m\angle L = 38^\circ$ (second largest)
   • $m\angle K = 22^\circ$ (smallest angle)

3. Relationship Between Angles and Sides:

   • The side opposite the largest angle is the longest.
   • The side opposite the smallest angle is the shortest.

   Therefore:

   • Side $KL$ (opposite $\angle M$) is the longest.
   • Side $MK$ (opposite $\angle L$) is medium.
   • Side $LM$ (opposite $\angle K$) is the shortest.

4. Correct Answer: The true statement is: $\boxed{\text{LM < MK < KL}}$

Let me know if you would like further clarification or detailed explanations.

5 Related Questions:

1. How can we determine the longest side of a triangle if given all its angles?
2. Why is the sum of the angles in a triangle always $180^\circ$?
3. How do we use the triangle inequality theorem to compare side lengths?
4. Can two angles in a triangle ever be equal if one is $120^\circ$?
5. How do side lengths relate to angles in obtuse triangles?

Tip:

Always remember that in any triangle, the side opposite the largest angle will always be the longest!