Question

given ( np cong mp ), ( np perp pl )
prove: ( \triangle npl cong \triangle mpl )
image of triangle with n, p, m on one side and l as the vertex, with right angles at p
statements reasons

1. ( np = mp ) given; ( np perp pl )
2. blank 2. def. of congruence
3. blank 3. definition of perpendicular
4. blank 4. all right angles are congruent
5. blank 5. reflexive property of congruence
6. blank 6. sas

Explanation:

Step1: Analyze Given Information

We know \( NP = MP \) (given), \( NP\perp PL \), and we need to prove \( \triangle NPL\cong\triangle MPL \). Let's fill the proof table step - by - step.

Step2: Statement 2 (Def. of Congruence)

If \( NP = MP \), by the definition of congruence (segments with equal length are congruent), we can say \( \overline{NP}\cong\overline{MP} \).

Step3: Statement 3 (Definition of Perpendicular)

Since \( NP\perp PL \), by the definition of perpendicular lines, the angles formed are right angles. So \( \angle NPL = 90^{\circ}\) and \( \angle MPL = 90^{\circ}\) (because \( NP\) and \( MP\) are on the same line and \( PL\) is perpendicular to \( NP\), so it's also perpendicular to \( MP\)). In terms of congruence, \( \angle NPL\cong\angle MPL \).

Step4: Statement 4 (All right angles are congruent)

We already established that \( \angle NPL\) and \( \angle MPL\) are right angles. By the property that all right angles are congruent, we confirm \( \angle NPL\cong\angle MPL \) (this is just re - emphasizing using the given reason).

Step5: Statement 5 (Reflexive Property of Congruence)

The reflexive property of congruence for a segment states that a segment is congruent to itself. So \( \overline{PL}\cong\overline{PL} \).

Step6: Statement 6 (SAS)

To use the SAS (Side - Angle - Side) congruence criterion, we need two sides and the included angle of one triangle to be congruent to the corresponding two sides and included angle of the other triangle. We have \( \overline{NP}\cong\overline{MP}\) (Statement 2), \( \angle NPL\cong\angle MPL\) (Statement 3 or 4), and \( \overline{PL}\cong\overline{PL}\) (Statement 5). So by SAS, \( \triangle NPL\cong\triangle MPL \).

Filling the table:

Statements	Reasons
2. \( \overline{NP}\cong\overline{MP} \)	2. Def. of Congruence
3. \( \angle NPL\cong\angle MPL \) (both are right angles)	3. Definition of Perpendicular
4. \( \angle NPL\cong\angle MPL \)	4. All right angles are congruent
5. \( \overline{PL}\cong\overline{PL} \)	5. Reflexive Property of Congruence
6. \( \triangle NPL\cong\triangle MPL \)	6. SAS

Answer:

The completed proof shows that \( \triangle NPL\cong\triangle MPL \) by SAS congruence criterion, with the steps as filled in the table above. The key congruent parts are \( \overline{NP}\cong\overline{MP}\), \( \angle NPL\cong\angle MPL\), and \( \overline{PL}\cong\overline{PL}\).