Question

in the figure shown, 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
point m is the midpoint of (overline{ln}) and (overline{pq}).
(qm cong mp)
(nm cong lm)
(angle qml cong angle pmn)
(\triangle lmq cong \triangle nmp)
reasoning
given
answer
definition of midpoint
answer
answer
answer
vertical angles
corresponding angles
ssa
segment fe is congruent to segment ec
sas
definition of midpoint

Explanation:

Step1: Analyze \( \overline{QM} \cong \overline{MP} \)

Since \( M \) is the midpoint of \( \overline{PQ} \) (given that \( M \) is the midpoint of \( \overline{LN} \) and \( \overline{PQ} \), but for \( \overline{PQ} \), midpoint means \( QM = MP \), so the reasoning is "Definition of Midpoint" (as midpoint divides a segment into two congruent parts).

Step2: Analyze \( \overline{NM} \cong \overline{LM} \)

Given \( M \) is the midpoint of \( \overline{LN} \), by "Definition of Midpoint", \( NM = LM \), so \( \overline{NM} \cong \overline{LM} \).

Step3: Analyze \( \angle QML \cong \angle PMN \)

\( \angle QML \) and \( \angle PMN \) are vertical angles, so by "Vertical Angles" theorem, they are congruent.

Step4: Prove \( \triangle LMQ \cong \triangle NMP \)

We have \( \overline{QM} \cong \overline{MP} \), \( \overline{NM} \cong \overline{LM} \), and \( \angle QML \cong \angle PMN \). By the "SAS" (Side - Angle - Side) congruence criterion (two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle), \( \triangle LMQ \cong \triangle NMP \).

Answer:

• For \( \overline{QM} \cong \overline{MP} \): Reasoning is "Definition of Midpoint"
• For \( \overline{NM} \cong \overline{LM} \): Reasoning is "Definition of Midpoint"
• For \( \angle QML \cong \angle PMN \): Reasoning is "Vertical Angles"
• For \( \triangle LMQ \cong \triangle NMP \): Reasoning is "SAS"