PQR is an isosceles triangle which PQ = PR

M and N are points on PQ and PR such that angle MRQ = angle NQR.

prove that triangles QNR and RMQ are congruent.

∠NQR, ∠NRQ and side QR for ΔQNR is equivalent to ∠MRQ, ∠MQR and side QR, which defines the two triangles and proves that the two triangles are congruent

Step-by-step explanation:

Here, we have

ΔPQR is an isosceles triangle,

∠MRQ = ∠NQR

Therefore, since ∡R = ∡Q (Base angles of isosceles ΔPQR), then;

∠MRQ + ∡Q + ∠QMR = 180° = ∠NQR + ∡R + ∠RNQ (Sum of angles in a triangle)

Whereby ∠MRQ = ∠NQR and ∡R = ∡Q, we have;

∠MRQ + ∡Q  = ∠NQR + ∡R which gives;

∠QMR = ∠RNQ

Which gives;

∠NQR, ∠NRQ and side QR for ΔQNR is equivalent to ∠MRQ, ∠MQR and side QR, which defines the two triangles and proves that the two triangles are congruent.