Question

Triangle LMN is a triangle in which altitude MP and NQ to
sides LN and LM respectively are equal. Show that
triangle LMP is congruent to triangle LNQ and prove that LM =LN​

Answer 1

Answer: For understanding this solution you should first know all the congruence rules...and CPCT...

Step-by-step explanation:In ΔMPN and ΔNQM

∠MPN  = ∠NQM  [ 90° EACH ]

MN  =  MN  [ COMMON ]

MP = QN  [ GIVEN ]

ΔMPN is congruent to  ΔNQM  [ RHS CONGRUENCY ]

⇒ ∠N = ∠M  [ CPCT ]

⇒ PN = QM  [CPCT ]  ...........(1)

since, ∠N = ∠M  

⇒ LM = LN  [ sides opposite to equal angles are always equal ]

Now, LM = LN  

subtracting QM from both sides, we get  

LM - QM = LN - QM

LM - QM = LN - PN  [ using (1) ]  

LQ = LP  ................(2)  

 

In Δ LMP and ΔLNQ

LM = LN  [ proved above ]

MP = NQ  [ given ]

LP  = LQ  [ using (2) ]

Δ LMP is congruent to  ΔLNQ  [ SSS ]

Hope it'll help you...