Question

in the figure above, point m is the midpoint of \\(\overline{ln}\\) and \\(\overline{pq}\\). use the given information to prove \\(\triangle lmq \cong \triangle nmp\\). move the correct answer box to each box. each answer may be used more than once. not all answers will be used.

statement	reasoning
\\(qm \cong mp\\)	answer
\\(nm \cong lm\\)	definition of midpoint
\\(\angle qml \cong \angle pmn\\)	answer
\\(\triangle lmq \cong \triangle nmp\\)	answer

\\(\text{sas}\\), \\(\text{corresponding angles}\\), \\(\text{segment fe is congruent to segment ec}\\), \\(\text{vertical angles}\\), \\(\text{ssa}\\), \\(\text{definition of midpoint}\\)

Explanation:

Step1: Analyze \( QM \cong MP \)

Since \( M \) is the midpoint of \( PQ \) (given), by the definition of midpoint, a midpoint divides a segment into two congruent segments. So the reasoning for \( QM \cong MP \) is "Definition of Midpoint".

Step2: Analyze \( \angle QML \cong \angle PMN \)

\( \angle QML \) and \( \angle PMN \) are vertical angles. Vertical angles are congruent, so the reasoning for \( \angle QML \cong \angle PMN \) is "Vertical Angles".

Step3: Analyze \( \triangle LMQ \cong \triangle NMP \)

We have \( QM \cong MP \) (from step1), \( \angle QML \cong \angle PMN \) (from step2), and \( NM \cong LM \) (from the statement, by definition of midpoint as \( M \) is midpoint of \( LN \)). So by the SAS (Side - Angle - Side) congruence criterion, \( \triangle LMQ \cong \triangle NMP \). So the reasoning for \( \triangle LMQ \cong \triangle NMP \) is "SAS".

Answer:

• For \( QM \cong MP \): Definition of Midpoint
• For \( \angle QML \cong \angle PMN \): Vertical Angles
• For \( \triangle LMQ \cong \triangle NMP \): SAS