To determine which congruence theorem can be used to prove that triangles \( \triangle MLQ \) and \( \triangle NPQ \) are congruent, let's analyze the information provided:
- Given: \( MQ = NQ \) (these sides are equal).
- Given: \( Q \) is the midpoint of \( LP\), which means \( LQ \) is equal to \( PQ \) (i.e., \( LQ = PQ \)).
- Given: \( LM \cong PN \) (these sides are congruent).
From the information above, we can note the following:
- We have two pairs of sides that are known to be equal: \( MQ = NQ \) and \( LM \cong PN \).
- Since \( Q \) is the midpoint of \( LP \), we have an additional pair of equal segments: \( LQ = PQ \).
Now, we can summarize:
- \( MQ = NQ \) (1 pair of sides)
- \( LM \cong PN \) (another pair of sides)
- \( LQ = PQ \) (yet another pair of sides)
Given these three pairs of equal (or congruent) sides, we can utilize the SSS (Side-Side-Side) congruence theorem, which states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
Thus, the answer is SSS.