In the diagram, if $\angle PQR = 48^\circ$ , what is the measure of $\angle PMN$?

Since  QR  =  PR  ,   △PQR  is isoscelese and its base angles are congruent. So...

 

∠PQR   =   ∠RPQ   =   48°

 

And vertical angles are congruent, so...

 

∠RPQ   =   ∠MPN   =   48°

 

Since  MP  =  NP  ,   △MNP  is isoscelese and its base angles are congruent. So...

 

∠PNM   =   ∠PMN

 

Since there are 180°  in every triangle...

 

∠MPN + ∠PMN + ∠PNM  =  180°

                                                         Plug in  48°  for  ∠MPN  and plug in  ∠PMN  for  ∠PNM

48° + ∠PMN + ∠PMN  =  180°

                                                         Combine like terms.

48° + 2(∠PMN)  =  180°

                                                         Subtract  48°  from both sides.

2(∠PMN)  =  132°

                                                         Divide both sides by  2 .

∠PMN  =  66°