Question

Given: MQ = NQ; Q is the midpoint of LP; LM ≅ PN

Triangles M L Q and N P Q are connected at point Q. A line is drawn from points M to N to form triangle M N Q.

Which congruence theorem can be used to prove △MLQ ≅ △NPQ?

AAS
SSS
ASA
SAS

Answer 1

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:

1. **Given**: \( MQ = NQ \) (these sides are equal).
2. **Given**: \( Q \) is the midpoint of \( LP\), which means \( LQ \) is equal to \( PQ \) (i.e., \( LQ = PQ \)).
3. **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**.