Question

example 3 proof write the specified type of proof. 11. two - column proof given: $np = mp$, $overline{np}perpoverline{pl}$ prove: $\triangle nplcong\triangle mpl$

Explanation:

Step1: State the given information

Statements	Reasons
\(\overline{NP}\perp\overline{PL}\)	Given

Step2: Define right angles

Since \(\overline{NP}\perp\overline{PL}\), \(\angle NPL = 90^{\circ}\). Also, if we assume \(\overline{MP}\) and \(\overline{PL}\) relationship (from the figure and given \(\overline{NP}\perp\overline{PL}\) and \(NP = MP\), we can say \(\angle MPL=90^{\circ}\) (because of the geometric construction). So \(\angle NPL=\angle MPL\) (all right angles are equal).

Statements	Reasons
\(\overline{NP}\perp\overline{PL}\)	Given
\(\angle NPL = 90^{\circ},\angle MPL = 90^{\circ}\)	Definition of perpendicular lines
\(\angle NPL=\angle MPL\)	All right - angles are equal

Step3: Use the common side

\(\overline{PL}\) is common to both \(\triangle NPL\) and \(\triangle MPL\). So \(PL = PL\) (reflexive property of equality).

Statements	Reasons
\(\overline{NP}\perp\overline{PL}\)	Given
\(\angle NPL = 90^{\circ},\angle MPL = 90^{\circ}\)	Definition of perpendicular lines
\(\angle NPL=\angle MPL\)	All right - angles are equal
\(PL = PL\)	Reflexive property of equality

Step4: Apply the SAS (Side - Angle - Side) congruence criterion

In \(\triangle NPL\) and \(\triangle MPL\), we have two sides and the included angle equal. That is \(NP = MP\), \(\angle NPL=\angle MPL\), and \(PL = PL\).

Statements	Reasons
\(\overline{NP}\perp\overline{PL}\)	Given
\(\angle NPL = 90^{\circ},\angle MPL = 90^{\circ}\)	Definition of perpendicular lines
\(\angle NPL=\angle MPL\)	All right - angles are equal
\(PL = PL\)	Reflexive property of equality
\(\triangle NPL\cong\triangle MPL\)	SAS (Side - Angle - Side) congruence criterion

Answer:

Statements	Reasons
\(\overline{NP}\perp\overline{PL}\)	Given
\(\angle NPL = 90^{\circ},\angle MPL = 90^{\circ}\)	Definition of perpendicular lines
\(\angle NPL=\angle MPL\)	All right - angles are equal
\(PL = PL\)	Reflexive property of equality
\(\triangle NPL\cong\triangle MPL\)	SAS (Side - Angle - Side) congruence criterion