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

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In the diagram,  if \$\angle PQR = 48^\circ\$ , what is the measure of \$\angle PMN\$? Mar 21, 2018

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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°

Mar 21, 2018