Question

1. examine the given figure. ( overline{np} ) bisects ( overline{lm} ) and ( angle m cong angle l ).

if ( overline{np} ) is a perpendicular bisector to ( overline{lm} ), then
( \triangle mpn cong \triangle lpn ) by ssa
( \triangle mpn cong \triangle lpn ) by sss
( \triangle mpn cong \triangle lpn ) by asa
( \triangle mpn cong \triangle lpn ) by aas
clear all

Explanation:

Step1: Analyze the given information

We know that \(NP\) bisects \(LM\), so \(MP = LP\). Also, \(\angle M\cong\angle L\) and \(\angle MPN=\angle LPN = 90^{\circ}\) (since \(NP\) is a perpendicular bisector).

Step2: Recall triangle congruence criteria

• SSA: Not a valid congruence criterion in general.
• SSS: Requires three pairs of equal sides. Here we have only one pair of sides (\(MP = LP\)) given from the bisector, and no information about the third side of each triangle (\(MN\) and \(LN\)) being equal.
• ASA: Requires two angles and the included side. The side is not included between the two angles here.
• AAS: We have two angles (\(\angle M\cong\angle L\) and \(\angle MPN\cong\angle LPN\)) and a non - included side (\(NP = NP\) by reflexivity).

Answer:

\(\triangle MPN\cong\triangle LPN\) by \(AAS\)