Question

point m is the midpoint of (overline{pq}), and (overleftrightarrow{lm}) is the perpendicular bisector of (overline{pq}). write a two - column proof to show that (lp = lq).
statements

1. (pm = qm), (overline{lm}perpoverline{pq})
2. (overline{lm}=overline{lm})
3. ()
4. (\triangle pmlcong\triangle qml)
5. ()
6. (lp = lq)

reasons

1. given
2. ()
3. right angle congruence theorem
4. ()
5. corresponding parts of congruent triangles are congruent.
6. ()

(the left side has some congruence and theorem labels like (overline{lm}congoverline{lm}), (overline{pm}congoverline{qm}), (angle pmlcongangle qml), (angle lmpcongangle lmq), (overline{lp}congoverline{lq}), (\triangle pmlcong\triangle qml), (\triangle lmpcong\triangle lmq), reflexive property of congruence, sas congruence theorem, hl congruence theorem, etc.)

Explanation:

Step1: Analyze Given Information

We know \( M \) is the midpoint of \( \overline{PQ} \), so \( PM = QM \) (definition of midpoint), and \( \overleftrightarrow{LM} \perp \overline{PQ} \), so \( \angle LMP = \angle LMQ = 90^\circ \) (definition of perpendicular lines). Also, \( \overline{LM} \) is common to both \( \triangle LMP \) and \( \triangle LMQ \).

Step2: Identify Triangle Congruence

In \( \triangle LMP \) and \( \triangle LMQ \):

• \( PM = QM \) (midpoint definition)
• \( \angle LMP = \angle LMQ = 90^\circ \) (perpendicularity)
• \( LM = LM \) (reflexive property of congruence)

By the SAS (Side - Angle - Side) Congruence Theorem, \( \triangle LMP \cong \triangle LMQ \).

Step3: Conclude Corresponding Parts

Since \( \triangle LMP \cong \triangle LMQ \), their corresponding parts \( LP \) and \( LQ \) are congruent (Corresponding Parts of Congruent Triangles are Congruent, CPCTC). So \( LP = LQ \).

Now, filling the two - column proof:

Statements	Reasons
2. \( LM = LM \)	2. Reflexive Property of Congruence
3. \( \angle LMP=\angle LMQ = 90^\circ \)	3. Definition of Perpendicular Lines
4. \( \triangle LMP\cong\triangle LMQ \)	4. SAS Congruence Theorem ( \( PM = QM \), \( \angle LMP=\angle LMQ \), \( LM = LM \))
5. \( LP = LQ \)	5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Answer:

The two - column proof is filled as above, and we conclude that \( LP = LQ \) by showing \( \triangle LMP\cong\triangle LMQ \) using SAS and then applying CPCTC.