Congruence of triangle

Question

In triangle PQR, PQ=QR and QM and RN are the bisectors of angle Q and angle R.  Prove that triangle NQR is congruent to triangle  MRQ

Answer 1

This is a false statement. The triangles NQR and MRQ cannot be congruent.

 

The original triangle PQR is an isosceles triangle and one of the triangles, MRQ is one of the mirror half-s of the original triangle. The other triangle, NQR is triangle created by bisecting one of the legs and one of the sides RN is the angular bisector to the leg.

 

For that reason, the triangles NQR and MRQ cannot be congruent. In the general case, the angular bisectors QM and RN have different lengths.