Question

10 examine the given figure. ( np ) bisects ( lm ) and ( angle m cong angle i ).

if ( np ) is a perpendicular bisector to ( lm ), then

( \triangle mpn cong \triangle lpn ) by ssa

( \triangle mpn cong \triangle lpn ) by asa

( \triangle mpn cong \triangle lpn ) by aas

( \triangle mpn cong \triangle lpn ) by sss

clear all

Explanation:

• Since \(NP\) is a perpendicular bisector of \(LM\), we have \(MP = LP\) (by the definition of a bisector).
• \(\angle MPN=\angle LPN = 90^{\circ}\) (by the definition of a perpendicular bisector).
• \(NP\) is common to both \(\triangle MPN\) and \(\triangle LPN\).
• We know \(\angle M\cong\angle L\) (given).
• The AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
• In \(\triangle MPN\) and \(\triangle LPN\), we have \(\angle M\cong\angle L\), \(\angle MPN\cong\angle LPN\), and \(NP = NP\) (non - included side).

Answer:

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