In Figure given below, Q > R, PA is the bisector of QPR and PM

Your question and diagram seems to be still wrong. So, if your question is :

In ΔPQR,  Q >  R. If PA is the bisector of  QPR and PM  perp  QR.

Prove that ∠APM =(1/2) (  Q -  R )

Then the Solution is as follows:

 

Given: In ΔPQR,  Q >  R. If PA is the bisector of  QPR and PM  perp  QR.

To Prove: ∠APM =(1/2) (  Q -  R )

Proof: Since PA is the bisector of ∠P,we have,

∠APQ=(1/2) ∠P....................(i)

In right -angled triangle PMQ,we have,

∠Q+ ∠MPQ=90°

∠MPQ= 90°-∠Q...................(ii)

∴∠APM=∠APQ-∠MPQ

 

Hence, the result.

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