Question

proof:
statements\treasons

1. ?\t1 given
2. ?\t2 def of congruence
3. ?\t3. definition of perpendicular.
4. ?\t4. all right angles are congruent.
5. ?\t5. reflexive property of ≅
6. ?\t6. sas

keyboard help
∠mpl and ∠npl are rt. angles
np = mp, np ⊥ pl
\overline{np} ≅ \overline{mp}
pl ≅ pl
△npl ≅ △mpl

Explanation:

Step1: Analyze Statement 1

The first statement's reason is "Given", so the statement should be the given information. From the diagram (or context), the given is likely \( NP = MP \), \( NP \perp PL \) (or the initial given facts). Wait, looking at the bottom, the first fact is \( \angle MPL \) and \( \angle NPL \) are rt. angles? Wait, no, the left side has:

First row (bottom left): \( \angle MPL \) and \( \angle NPL \) are rt. angles

Second: \( NP = MP \), \( NP \perp PL \)

Third: \( \overline{NP} \cong \overline{MP} \)

Fourth: \( PL = PL \)

Fifth: \( \triangle NPL \cong \triangle MPL \)

So let's match with the proof table:

1. Statement 1: Reason is Given. So the given should be the initial info, like \( \angle MPL \) and \( \angle NPL \) are right angles, \( NP = MP \), \( NP \perp PL \)? Wait, no, let's go step by step.

Step2: Statement 2: Reason is Def of congruence. So if we have a segment congruence, like \( \overline{NP} \cong \overline{MP} \) because \( NP = MP \) (def of congruence: segments with equal length are congruent). So Statement 2: \( NP = MP \) (wait, no, def of congruence for segments: if \( NP = MP \), then \( \overline{NP} \cong \overline{MP} \)). Wait, maybe:

1. Statement 1: \( \angle MPL \) and \( \angle NPL \) are right angles (Reason: Given) – but wait, the bottom left first is \( \angle MPL \) and \( \angle NPL \) are rt. angles.

2. Statement 2: \( \overline{NP} \cong \overline{MP} \) (Reason: Def of congruence, since \( NP = MP \))

3. Statement 3: \( NP \perp PL \) (Reason: Definition of perpendicular, since they form right angles)

4. Statement 4: \( \angle MPL \cong \angle NPL \) (Reason: All right angles are congruent)

5. Statement 5: \( \overline{PL} \cong \overline{PL} \) (Reason: Reflexive Property of \( \cong \))

6. Statement 6: \( \triangle NPL \cong \triangle MPL \) (Reason: SAS, since we have two sides and included angle congruent: \( \overline{NP} \cong \overline{MP} \), \( \angle NPL \cong \angle MPL \), \( \overline{PL} \cong \overline{PL} \))

Let's fill each statement:

1. Statement 1: \( \angle MPL \) and \( \angle NPL \) are right angles (Reason: Given)

2. Statement 2: \( \overline{NP} \cong \overline{MP} \) (Reason: Def of congruence, because \( NP = MP \))

3. Statement 3: \( NP \perp PL \) (Reason: Definition of perpendicular, since \( \angle NPL \) is a right angle)

4. Statement 4: \( \angle MPL \cong \angle NPL \) (Reason: All right angles are congruent)

5. Statement 5: \( \overline{PL} \cong \overline{PL} \) (Reason: Reflexive Property of \( \cong \))

6. Statement 6: \( \triangle NPL \cong \triangle MPL \) (Reason: SAS)

Now let's check the bottom left:

First: \( \angle MPL \) and \( \angle NPL \) are rt. angles – matches Statement 1.

Second: \( NP = MP \), \( NP \perp PL \) – maybe Statement 1 and 3? Wait, no, let's list each statement with reason:

1. Statement: \( \angle MPL \) and \( \angle NPL \) are right angles; Reason: Given

2. Statement: \( \overline{NP} \cong \overline{MP} \); Reason: Def of congruence (since \( NP = MP \))

3. Statement: \( NP \perp PL \); Reason: Definition of perpendicular (since \( \angle NPL \) is a right angle)

4. Statement: \( \angle MPL \cong \angle NPL \); Reason: All right angles are congruent

5. Statement: \( \overline{PL} \cong \overline{PL} \); Reason: Reflexive Property of \( \cong \)

6. Statement: \( \triangle NPL \cong \triangle MPL \); Reason: SAS

So filling the table:

1. Statement: \( \angle MPL \) and \( \angle NPL \) are right angles

2. Statement: \( \overline{NP} \cong \overline{MP} \)

3. Statement: \( NP \perp PL \)

4. Statement: \( \angle MPL \cong \angle NPL \)

5. Statement: \( \overline{PL} \cong \overline{PL} \)

6. Statement: \( \triangle NPL \cong \triangle MPL \)

Let's verify each reason:

1. Given: Correct, as it's the initial info.

2. Def of congruence: Correct, since \( NP = MP \) implies segment congruence.

3. Definition of perpendicular: Correct, since perpendicular lines form right angles.

4. All right angles are congruent: Correct, so \( \angle MPL \cong \angle NPL \).

5. Reflexive Property of \( \cong \): Correct, a segment is congruent to itself.

6. SAS: Correct, two sides (\( \overline{NP} \cong \overline{MP} \), \( \overline{PL} \cong \overline{PL} \)) and included angle (\( \angle NPL \cong \angle MPL \)) congruent.

Answer:

1. Statement: \( \angle MPL \) and \( \angle NPL \) are right angles
2. Statement: \( \overline{NP} \cong \overline{MP} \)
3. Statement: \( NP \perp PL \)
4. Statement: \( \angle MPL \cong \angle NPL \)
5. Statement: \( \overline{PL} \cong \overline{PL} \)
6. Statement: \( \triangle NPL \cong \triangle MPL \)