Question

which postulate or theorem could be used to prove triangleLMQ is congruent to triangleLNP

Answer 1

The answer to this query is AA similarity postulate. 

Because the triangles given are only similar in angle but dissimilar in sides which makes it incongruent with respect to the sides, AA similarity postulate is the exact answer. 

SAS ASA are not possible answers. 

Answer 2

Measure each of your sides make sure all of your sides are even/similar (LMQ & LNP)

So, AA Similarity Postulate (option C)

The rest of the options are false to this problem!