(you) $ m $ is the midpoint of $ \overline{pq} $, and $ \overline{lm} $ is the perpendicular bisector of $ \overline{pq} $. write a two - column proof to show that $ lp = lq $.
statements
1. $ pm = qm $, $ lm \perp pq $
2. $ lm = lm $
3.
4. $ \triangle pml \cong \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.

(left side list of statements/reasons for selection: \( lm = lm $, $ lm \perp pq $, $ \angle lmp = \angle lmq $, $ lp = lq $, $ \angle lmp \cong \angle lmq $, $ \triangle pml \cong \triangle qml $, reflexive property of congruence, sas congruence theorem, hl congruence theorem, sss congruence theorem, definition of segment congruence)

Question

(you) $ m $ is the midpoint of $ \overline{pq} $, and $ \overline{lm} $ is the perpendicular bisector of $ \overline{pq} $. write a two - column proof to show that $ lp = lq $.
statements
1. $ pm = qm $, $ lm \perp pq $
2. $ lm = lm $
3.
4. $ \triangle pml \cong \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.

(left side list of statements/reasons for selection: \( lm = lm $, $ lm \perp pq $, $ \angle lmp = \angle lmq $, $ lp = lq $, $ \angle lmp \cong \angle lmq $, $ \triangle pml \cong \triangle qml $, reflexive property of congruence, sas congruence theorem, hl congruence theorem, sss congruence theorem, definition of segment congruence)

Accepted Answer

# Explanation:
## Step1: Analyze Given Information
We know $ \overline{PM} \cong \overline{QM} $ (given as $ PM = QM $ in statement 1), and $ \overline{LM} $ is the perpendicular bisector of $ \overline{PQ} $, so $ \angle PLM $ and $ \angle QLM $ are right angles (by definition of perpendicular bisector), and $ \overline{LM} \cong \overline{LM} $ (reflexive property).

## Step2: Identify Triangle Congruence
For $ \triangle PLM $ and $ \triangle QLM $, we have:
- $ \overline{PM} \cong \overline{QM} $ (given)
- $ \overline{LM} \cong \overline{LM} $ (reflexive)
- $ \angle PLM \cong \angle QLM = 90^\circ $ (perpendicular bisector)

By the Hypotenuse - Leg (HL) Congruence Theorem (since they are right triangles, hypotenuse $ PM = QM $ and leg $ LM = LM $), $ \triangle PLM \cong \triangle QLM $.

## Step3: Corresponding Parts of Congruent Triangles
Since $ \triangle PLM \cong \triangle QLM $, their corresponding parts $ \overline{LP} $ and $ \overline{LQ} $ are congruent (CPCTC: Corresponding Parts of Congruent Triangles are Congruent). So $ LP = LQ $.

Now, let's fill the two - column proof:

### Statements Column:
1. $ PM = QM $, $ \overline{LM} \perp \overline{PQ} $ (Given)
2. $ \angle PLM $ and $ \angle QLM $ are right angles (Definition of perpendicular bisector)
3. $ LM = LM $ (Reflexive Property of Congruence)
4. $ \triangle PLM \cong \triangle QLM $ (HL Congruence Theorem)
5. $ LP = LQ $ (CPCTC)

### Reasons Column:
1. Given
2. Definition of perpendicular bisector (A line perpendicular to a segment forms right angles with the segment)
3. Reflexive Property of Congruence (A segment is congruent to itself)
4. HL Congruence Theorem (If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, the triangles are congruent)
5. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

# Answer:
The two - column proof is filled as follows:

| Statements | Reasons |
| --- | --- |
| 1. $ PM = QM $, $ \overline{LM} \perp \overline{PQ} $ | 1. Given |
| 2. $ \angle PLM $ and $ \angle QLM $ are right angles | 2. Definition of perpendicular bisector |
| 3. $ LM = LM $ | 3. Reflexive Property of Congruence |
| 4. $ \triangle PLM \cong \triangle QLM $ | 4. HL Congruence Theorem |
| 5. $ LP = LQ $ | 5. CPCTC |

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze Given Information

Step2: Identify Triangle Congruence

Step3: Corresponding Parts of Congruent Triangles

Statements Column:

Reasons Column:

Answer:

Statements	Reasons
2. \( \angle PLM \) and \( \angle QLM \) are right angles	2. Definition of perpendicular bisector
3. \( LM = LM \)	3. Reflexive Property of Congruence
4. \( \triangle PLM \cong \triangle QLM \)	4. HL Congruence Theorem
5. \( LP = LQ \)	5. CPCTC