Question

example 3
proof write the specified type of proof.

1. two - column proof

given: ( np = mp ), ( overline{np} perp overline{pl} )
prove: ( \triangle npl cong \triangle mpl )

1. two - column proof

given: ( ba = cd ), ( overline{ab} parallel overline{cd} )
prove: ( \triangle acd cong \triangle dba )

Explanation:

Step1: List the given information

Statements	Reasons
\(\overline{NP}\perp\overline{PL}\)	Given
\(\angle NPL=\angle MPL = 90^{\circ}\)	Definition of perpendicular lines
\(PL = PL\)	Reflexive property of equality

Step2: Apply the SAS (Side - Angle - Side) congruence criterion

Since we have two sides and the included angle of \(\triangle NPL\) and \(\triangle MPL\) equal (\(NP = MP\), \(\angle NPL=\angle MPL\), \(PL = PL\)), by the SAS congruence criterion, \(\triangle NPL\cong\triangle MPL\).

Answer:

Statements	Reasons
\(\overline{NP}\perp\overline{PL}\)	Given
\(\angle NPL=\angle MPL = 90^{\circ}\)	Definition of perpendicular lines
\(PL = PL\)	Reflexive property of equality
\(\triangle NPL\cong\triangle MPL\)	SAS (\(NP = MP\), \(\angle NPL=\angle MPL\), \(PL = PL\))